Michael Greinecker's answer is correct. But there is another subtler weaker sense in which what you asked for could be true.
Namely, the method of forcing shows that every set that exists in any model of set theory can become countable in another larger model of set theory, the forcing extension obtained by collapsing the cardinality of that set to ω. Thus, the concept of countability loses its absolute meaning; whether a set is countable or not depends on the set theoretic background.
So if X is any set, then in some forcing extension of the universe, the set X becomes countable.
In particular, this will be true when X is itself a model of set theory.
There are various ways of viewing the nature of existence of these forcing extensions, among them Boolean-valued models and their quotients, the Boolean ultrapowers, and I refer you to the set-theoretic literature. If one uses the Boolean ultrapower, then stating the theorem from inside V, one attractive way to describe the situation is as follows:
Theorem. If V is the universe of sets and X is any particular set, then there is a class model (W,E) of set theory, and an elementary embedding of V into a submodel W0 of W, such that the image of X in W is thought by W to be countable.
Basically, the model W0 is the Boolean ultrapower of V by the forcing B to collapse X, using any ultrafilter on B, and W is the quotient of the Boolean valued model VB. The elementary embedding maps every object y to the equivalence class of the check name of y.