First-order logic is complete. There are no counterexamples. Since the Godel sentence of a theory is not provable from the axioms of a theory, it follows (from completeness) that the Godel sentence must be false in some "non-standard" models of the axioms.

You can read the first incompleteness theorem as saying that there's no way to come up with a finitely expressible set of axioms that can prove all the truths of arithmetic. This is not in conflict with the completeness theorem. The completeness theorem tells us that if there were some finite (or recursively enumerable) set of axioms of arithmetic such that the inference to any truth of arithmetic from those axioms was semantically valid, then all truths of arithmetic can be proven from that set of axioms. The incompleteness theorem tells us that the antecedent of this conditional is false; it doesn't deny the truth of the conditional itself.