Alligators do not hibernate during winter, but they do brummate. Brummate is when an alligator slows down its metabolic rate, but doesn't enter a state of hibernation.
Alligators are cold-blooded animals and prefer warm weather and warm water temperatures. In areas where winters are cold, alligators will try to stay submerged in water or find a burrow to rest. These areas provide less temperature changes than being out in the open air and can help the alligator retain warmth.
When alligators brummate they remain very still. Many will lie near the bottom of their burrow, only surfacing to breathe. Some float on top the water or lie on vegetation during the coldest parts of winter.