Birds, fish, reptiles, amphibians, insects? Sure. But no mammals.
So I had to google it. Apparently, there is a sloth that moves around so slowly moss grows all over it and it doesn't care. So it may appear green, but only in the sense that it wears it.
This is an interesting question but I don't think it is restricted to green. Isn't the same true of purple, blue and red? I'm not talking about just reddish like human hair or a red panda but truly bright red like a cardinal. I would imagine it has something to do with our evolutionary history. Complete speculation here but laced with a few facts I picked up. I hear the common ancestor of mammals emerged around the time the dinosaurs became extinct and was basically a tiny rodent like a shrew. I wonder if as a small animal that can't fly or swim it had to hide a lot and basically just came in shades of brown. So maybe any genes for other colors were lost before that common mammalian ancestor emerged and although mammals have lots of patterns they don't have many colors.