This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
The alligator (genus Alligator) was an amphibious carnivorous reptile indigenous to Earth. The American alligator's large and powerful jaws made it a danger to Humans in the southern United States of America.
Rain Robinson had a snow globe cradled by an alligator figure on her desk at Griffith Observatory in 1996. When time-traveling Tom Paris and Tuvok broke in that office that year, Paris studied it with some fascination. (VOY: "Future's End")
While being interrogated by Dolim, a Xindi-Reptilian, who proclaimed his cold-blooded superiority, Jonathan Archer attempted to provoke him by recalling the downfall of reptilian rule on Earth, and the rise of the mammals. Archer further explained that not all reptiles died out, and that some evolved into snakes, alligators and turtles. (ENT: "Azati Prime")
Joseph Sisko kept a stuffed alligator in his restaurant, Sisko's, in New Orleans, hanging it from his ceiling. When Jake Sisko was younger and visited, Joseph would tell him that it was "only in stasis, and at night it came down and guarded the restaurant". He later joked that he stopped doing that when it "got to much of a hassle to wrestle it back up to the ceiling." (DS9: "Homefront")