Ha! What a great thought. Alas, there is no evidence.

Alligators are indigenous to the U.S. and China, so medieval Europe did not know of their existence. They did know about crocodiles, though. The early medieval scholar Isidore of Seville included them in his encyclopedic work of knowledge, the Etymologies, and so later medieval bestiaries mentioned them as well. These snippets informed their readers that crocodiles live in the Nile and eat people while crying (hence, I guess, the expression "crocodile tears"?).

However, it's pretty clear that this knowledge was only second and third and fourth-hand, repeated from Isidore drawing on classical sources. Some late medieval texts refer to crocodiles as hailing from the land of the mythical Prester John (a rumored Christian king in pagan lands), which basically means the crocodile had entered the realm of mythology. Furthermore, manuscript illuminations of crocodiles range from stegosaurus-like to rather like an oversized hunting dog.

And at any rate, an 'exotic' beast like that in Europe would have to be a major gift from an African ruler to a European one. It would certainly not end up in a moat to threaten potential invaders. It would earn prize of place in a royal menagerie for sure. (And that's assuming a perfect history of being able to be caught and transported safely and dealing with the changing weather conditions and, one supposes, limited food sources available in a castle moat.)