The alligator aspect to the name is that the bark is surely gnarled and resembles the skin of an alligator. I’ve added a photo of our tree. It displays the bark quite clearly. The tree in the photo is a bit deceptive, I think. It is a large tree, but not as large as it appears in the photo. The diameter is perhaps about 18-20 inches. And I had a closer look at the height; it is around 25 feet tall. Alligator junipers usually range from about 32 to 50 feet tall at maturity.
As a matter of interest, we have a college in our small city; the name is Prescott College. They have a unique publication that is published only once per year. The name of their magazine is the Alligator Juniper. Check it out. It looks interesting.