This is the Legendary Alligator location also known as the Legendary Bullgator.
You can't hunt the Legendary Alligator until you encounter it in the story mission, "Country Pursuits", in chapter 4. Some people have not been able to get it to appear until the beginning of Chapter 6.
It is quick and can take quite a few shots. Use your sniper rifle from far away and then switch to your shotgun and use Dead Eye when it gets close.
If you can't kill it before it gets to you, you can run away until it stops chasing you then shoot it again while it's running away.