In 2006, Gao and Zeng gave the free field realizations of highest weight modules over the extended affine Lie algebras gl_2(C_q) for any nonzero complex number q. In this talk, applying the technique of localization to those free field realizations, we construct a class of new weight modules with infinite dimensional weight multiplicities. We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite dimensional weight multiplicities over gl_2(C_q) for any nonzero q.