If ZFC has a model it would have to be infinite. This can follow, as said from power set or the infinity axiom. Furthermore the language of set theory has only one binary relation $\in$, so any theory would be countable and therefore if there is an infinite model there would have to be a countably infinite model.
All this was said before, but I would like to add on an important point:
Even if $\frak M$ is a countable model of ZFC, internally it is a proper class. That is to say, there is no $f\in\frak M$ such that $f$ is a bijection between $\omega$ and $\frak M$.
This model, along with a function witnessing its countability live in a larger model of some strong-enough-theory (this larger model may be a class model).
Note that this has nothing to do with countability. Every set-model of ZFC would think of itself as a proper class, but we "know" (externally) that it is only a set, and if this set happens to live in a universe of ZFC then there is some function from an ordinal (which may be an element of this set-model) onto that model. This should be a hint of how complicated and convoluted infinite objects can get.