First of all, the phrasing is a little odd. A model does not "say" anything: a sentence in a first-order theory says something about a variable, and the sentence is satisfied or not by substitution of elements from the universe of the model for that variable. Given that the universe of the model cannot be an element of itself, the sentence cannot say anything about the universe of the model, much less of the model itself. Second-order logic will also not allow the formula to say anything about the universe, because although the variables can now range over subsets of the universe, the universe is a proper class, not a subset. So, in order to say anything about this model, you need to use a new universe which has the old universe as a subset, and the appropriate relations in the new universe. And then you can have a theory containing sentences which say that the old universe is countable.
Countability is not an absolute notion.