You usually don't see goats in nativity scenes, live or otherwise. This NY Times item even reports of a goat escaping from a live nativity scene. For that matter, goats don't usually show up in Christian iconography except in Last Judgement scenes, in which they represent the damned.
Having lived around them, I'd have to say that they don't seem to have much of an affinity for Christianity either. In fact, I detect a certain fondness for classical Greco-Roman paganism among them. Zeus, father of gods and men, was after all nursed by a goat as an infant on the island of Crete and they were also associated with Dionysus.
Maybe that explains why our goats seem more into Saturnalia than Christmas at this time of year.