Today we have a religious spread question! I know that the Aztecs conquered their nearby neighbors, but did they convert them to their beliefs as well? This was sort of brought on by the Ainu question... Well, this is a complicated and weird question to answer, because there're a lot of factors at work, but we'll take a crack at it! (Most romantic Valentine question ever, I know.)
First of all, a quick terminology note: while most of us here in the west refer to the dominant kingdom at the time of the Mexican invasion as "Aztecs", in fact they did not ever call themselves that. The word means "people from Aztlan", with Aztlan being a mythical land that they believed their ancestors were first born in and from which they had traveled to found their empire far away, and while they did use it occasionally to refer to members of different ethnic groups that claimed descent from the people of Aztlan, it was never a word used to casually refer to themselves. As usually happens with these things, a nineteenth-century Prussian dude started using the word to refer to all the peoples in the area as a general whole, and then other white dudes jumped on the bandwagon because it gave them a convenient way to label the pre-conquest "savages" as "Aztec" and the post-conquest "civilized" people as "Mexican", and the rest is history.
So, generally, we try to talk about the Mexica instead of the Aztecs - the Mexica being the indigenous ethnic group that founded Tenochtitlan and was ruling the area at the time of the European invasion, and whose mythology is populated by such delightful characters as the warrior sun god Huitzilopochtli and the jaguar-spotted bringer of misfortune Tezcatlipoca. Different ethnic groups in what is now Mexico had their own mythologies and cultural features that can get lost when they get mixed into the general pot of "Aztec", and 95% of the time when folks talk about the Aztecs, they really mean the Mexica specifically anyway, so we try to make that distinction for them where possible. (There are moments where it is more appropriate, usually when discussing the Triple Alliance of cities in the empire, since not all of them were Mexica cities and territories, or the language branch, which is still referred to as Uto-Aztecan, so you may see that once in a while, though!)
Anyway, on to our question about proselytizing Mexicans. You are correct that the Mexica, once they managed to establish themselves as a power in central Mexico, did indeed set about conquering everyone they could come into easy contact with; prior to the founding of Tenochtitlan, they were essentially a people who spent most of their time being semi-nomadic mercenaries with no power or territory of their own, and once they got their first taste of structured power, they overcorrected in a big way. They came up from nothing, and so they believed that, in contrast to other peoples in the area who prized divine lineages or established dynasties, the only way they were going to get anything would be to make it absolutely clear that they were taking it. They saw their strength in warfare and conquest as direct confirmation of their right to rule - basically, if they could conquer someone, that was proof that they were supposed to, because it showed that they were more deserving of having power than the new conquerees.
And, as is pretty much always the case when it comes to a people with a strong unified religion, they tied that religion into their expansionism. Huitzilopochtli, the totem god of the Mexica, was also the god of martial prowess and victory, and so they were very easily able to say that from him came their power to conquer, and that their conquests were further proof of his power as well as their own. And since their god was so badass and clearly kicking all the divine ass there was to kick, it was easy to further reason that other peoples' gods were therefore just, like, way less cool.
Now, the Mexica were something of a baby empire, rocketing to power in central Mexico where there had been other gigantic and very influential empires (the Maya, the P'urepecha, the Olmec and so on) over the centuries, so they were heavily influenced by their neighbors and predecessors. Various deities and practices in their religion were closely related to those of other peoples in the area, which is why scholars find it easy to talk about features that all Mesoamerican religions tend to have in common, such as big goggle-eyed rain monsters or a reverential fondness for the ritual shedding of blood. That doesn't mean these religions were the same, of course - often they might share features in common but attach different interpretation or significance to them - but they had a lot of things in common, which meant that it was easy for them to be shared back and forth. The Mexica were not exempt from this sort of religious melting pot experience; they borrowed and traded deities and ideas as much as anyone else in the area, and perhaps did so even more so than most, since they were setting up a mighty empire in basically a couple of weeks in empire-building time, and they could use all the cultural support and material they could get.
(There's an entire dissertation, I'm sure, just in the Mexica interest in the culture of the Toltec people who preceded them, and the way that they modeled much of their own empire after it and claimed to be its spiritual inheritors, but there's enough going on in this post for now. Another time!)
So the Mexica were both heavily invested in their god being the coolest of all gods, and also in borrowing and making their own various other deities from cultures they encountered, which brings us around the the third facet of religious contact: encouraging (or, you know, forcing) other peoples to adhere to their religion. Some of this did in fact go on, but because Mexica religious thought was very different from what we generally think of as the European inquisition-style types of conquest conversion, it didn't come about in the same way. Certainly there was establishment of state religion in new parts of the empire, but there was not necessarily eradication of other religious traditions, especially when they could be considered just local variants on Mexica beliefs anyway, and the crowning form of "converting" people to the worship of the Mexica gods was simple: they just carried other peoples' gods away and said "these are ours now, so you have to be in our religion if you want to worship them."
Several different Mesoamerican religions include the concept of a sacred bundle; the Mexica called it the tlaquimilolli, and it was centrally important to the process of conquering and absorbing another culture's religion. The tlaquimilolli was a bundle of cloth that contained religiously significant items dedicated to a specific god, which might include items sacred to that god, a mask or statue of the deity in question, or offerings given to them in times gone by. The bundle was carried with the god's worshipers when they were traveling, or kept safe in a temple or holy location when they were not; it allowed the people to have a literal, concrete form of their god that could be touched and revered, without actually expecting the god to show up and lay down the law in person. The Mexica were extremely fond of the tlaquimilolli, which had allowed them to carry their gods with them when they were nomads, and which once they were established were seen as irrevocable proof that their gods had traveled with them and guided them to their new glory.
So I'm sure you all can see where I'm going with this: if the tlaquimilolli in a symbolic sense contains the god, and the god goes where the bundle goes, then all you have to do to transplant and assert power over the god is to literally steal that poor deity. A signature Mexica conquest move was to bust into the local temple of wherever they were conquering, find the tlaquimilolli of the resident deity, and then rush it back home to one of the official state temples to be housed with the rest of the gods, all the while yelling very official wartime variants on "look at your wussy god, we captured him because our gods are way better!" and "guess you know you lost the war now that we have your god!" and "he's going to go live with us where real gods go, you should start learning more about real gods!"
Which is a fairly passive-aggressive form of religious conversion, when you think about it, but the Mexica goal wasn't really to demand that everyone get on board with their exact religious practices. They just wanted to assert that they were the bosses and their gods were, too, and past that didn't much care if you continued to worship your obviously inferior gods, just as long as you participated appropriately in empire-wide religious observances and everyone understood that non-Mexica-approved gods clearly were inferior when compared to the glory of Huitzilopochtli and his cronies. And, like conquered peoples everywhere throughout history, many of the people they steamrollered nodded and smiled politely and then went right back to doing what they were doing, because if the Mexica weren't actually going to do anything about it and it wasn't going to cause any major problems, there was no reason not to. In fact, some peoples even just headed the Mexica off and placed decoy tlaquimilolli, allowing the Mexica to "capture" the god as far as they were aware but keeping the real bundles hidden safely away where they could be dusted off and remain with their people after the conquerors had mostly left.
It's no secret that religion often plays a key role in expansionism and warfare, either nominally or seriously, and that various cultures have attempted to convert those other people they conquer in order to unify their territories and head off dissent, but the form that such conversions take was often very different depending on the place, the history, and the religion in question. In Mexico, where half of every religion is a synthesis from three religions that came before it and a quarter of every pantheon may or may not also be in someone else's pantheon anyway, there was far less need for the kind of conquest that involved stamping out the local religion, and much more of a chance for conquerors to look at the religions they encountered, say, "eh, close enough as long as you remember who's boss," and just let them keep doing their thing.