The Apex Alligator fight is located in Fawtick Bayou. It's actually not the first boss fight, though it's the first region you unlock.
Alligators are tough to fight. With strong jaws they'll try to clamp down on you, which costs you a lot of health. Dodge with RB when they lunge for you, then attack from below or the side when possible.
When you defeat the Apex Alligator you'll unlock the Amphibious Organ mutation.