Killing alligators in Assassin’s Creed Liberation may prove to be extremely complicated at first. I swear the first time I encountered alligator in AC Liberation it took me like half an hour to kill the bastard.
There are no tricks to killing the alligators though. All you have to do is press the buttons at the right time. First one is always E, and then it will be one of these three: SPACE, Q or Left Mouse Button. A success is when you press E in time and them the second required button right after it appears, if you do it correctly they will light up in green. To kill an alligator you have to succeed two times in a row.
Don’t stress about it. Just wait for the button to appear on the screen and hit it. You will get how it’s done eventually. And once you’ve taken down several alligators, doing it won’t seem that hard anymore.
Mostly you can just avoid killing alligators in Assassin’s Creed Liberation altogether. However there is one alligator fight that is mission related and you will not be able to avoid it. You will also have to kill alligators if you’re going to collect the twelve alligator eggs, that are a part of the required collectibles for the Collector trophy. This can be avoided if you get Alligator Hunter Hat - it will stop the alligators from attacking you. However you have to get it as a reward on uplay for 30 units and this may not be what you want to do. After collecting all alligator eggs you will be awarded with special Traveler’s Hat which will do the very same thing. However I have no idea why anyone would go anywhere near the alligators after collecting the eggs.