Making ships that could navigate high seas (not hug a coast) is pretty difficult, and requires a well-developed maritime culture, which itself requires a well-developed maritime interest. The Mediterranean encouraged naval technology because useful goods were spread between disparate societies; the Nile is straightforwardly better for growing wheat than the Greek highlands are, while if you want quality copper you can't beat Cyprus. Moreover, some parts of the Mediterranean (Egypt and the Levant, ancient North Africa, then later Italy and southern France) have a high population carrying capacity, with no practical limit.
The Maya just didn't really have this. Pretty much all goods they really needed were produced in abundance in one part of the Maya sphere or another, so not much essential stuff ever passed through non-Maya hands. The elite stuff like jade did need a longer trade network, sure, but never one that's so debilitating you'd need to conquer anywhere else. And even then, that's all to the south! If Maya oarmen had somehow reached the Caribbean islands or the north shore of the Gulf, they'd have found a bunch of people who have basically nothing of use. There was no economic motive to develop or build ships that could do more than hug the coast.
But let's assume they did so. What would incentivise them to build a new city there? Classic Maya society was an extremely conservative aristocracy, where priests and nobles ruled over their own city-state and its thousands of commoners who themselves had neither the means or the motive to go anywhere else. The city wasn't just an important economic unit, it was the focal point of the entire religious and cultural system. There weren't any aristocrats on the Mayflower, and unlike 17th-century England the Maya aristocrats had no intention of allowing their subjects to leave - at least, they certainly wouldn't have financed the undertaking.
But let's assume they do for some reason. What are the native peoples who already live there going to think? The Maya would probably think of them as uncivilized, sure, but there wasn't exactly a massive disparity in technology, nor would they introduce any devastating diseases. Absolutely nothing about Maya material culture would have any use in the bayous or on the coastal plains. Our poor Maya "colonists" would find themselves astonished that their intimate knowledge of water conservation and building with stone don't do much for them in a swamp, and instead they'd have to learn how to build levees and mix their crops really quickly. So quickly, in fact, that I suspect we'd stop calling them "Maya" and instead call them "Mississippians" within two or three generations. Considering both of the above points, it's not like they'd have had a particularly close connection to their original homeland anyway.
People on this board take too much from the 17th and 18th centuries (as well as just plain hindsight) in thinking that "populous colonies = good". Even the European colonisation of the Americas started as trade enterprises, and some colonies like New France barely ever got beyond that level. Even Rome, the sine qua non of premodern colonisation, didn't really send out colonial expeditions - they preferred to conquer the land first, and then dole it out to Roman families later if they had to, or else just assimilate the natives. It should be a basic rule of thumb that settler colonisation just didn't really happen before the 18th century, and unfortunately I can't see the Maya becoming an exception.