The first few centuries of European presence in the Americas were marked by an intense interest in the "curiosities" and "natural wonders" of the New World, very much in the mode we might think of as antiquarian instead of natural history. That is, Europeans eagerly collected items and stories, maybe for private treasuries or organized for public collections, but from the standpoint of outside fascination rather than study to understand. Descriptions of the flora and especially the fauna of the Western Hemisphere fell into this category.

The elite literary tradition of medieval Europe did offer something vaguely similar to alligators: the crocodile. Medieval bestiaries (a kind of text that used descriptions of animals as moral allegories for humans to learn how to/how not to behave) described crocodiles as "a curse on four legs" that lives in the Nile and is equal parts deadly and hard to kill. Medieval artists' skills were not necessarily always up to snuff, but the general idea of a GIANT, TOOTHY LIZARD THAT WILL KILL YOU certainly existed in the medieval educated imagation.

I mention this background because at least one New World account, Francis Moore's journal from 1736 Georgia, made the connection to inform his future readers (he intended to, and did, publish his journal):

Of Reptiles, the Crocodile, which seems to be the chief, abounds in all the Rivers of Georgia; they call them Alligators.

However, Europeans did not as a general rule know crocodiles firsthand, only through books. As a result, we don't see as much comparison between crocodiles and alligators as we might in descriptions of animals like ducks and squirrels. (Comparisons that could go either way--some authors felt very strongly that the European version needed to be superior in all cases, but not all of them.)

The nature writers/cataloguers of the New World did not seek to turn their subjects into moral allegories like the Roman-medieval bestiary tradition; you'll find no discussions of false "alligator tears" (crocodile tears are tears for show instead of genuine remorse, an expression with its root in the bestiaries). They certainly had a pattern when discussing alligators, though: terrifying.

Hernandez de Soto described how alligators lie in wait to kill humans and animals. (His account also describes mosquito bites and snakes "the size of a man's thigh"--welcome to Florida). William Bartram, a naturalist in Revolutionary America, provided more colorful commentary:

Behold the alligator rushing forth from the flags and reeds. His enormous body swells...the waters like a cataract descend from his opening jaw...The earth trembles with his thunder...The alligators gathered around my harbor from all quarters...They stuck their jaws together so close to my ears as almost to stun me, and I expected every moment to be dragged out of the boat and instantly devoured.