You are referring to Goedel's Incompleteness Theorem (an extremely important development in mathematics almost a century ago which has very broad implications), but you mischaracterize its meaning somewhat.
The theorem basically states that no system of axioms can be both consistent and decideable. The nature of the system of axioms is arbitrary, and God may or may not be a part of it. One problem is that any consistent set of axioms has things that are not provable within that set of axioms, and that you can infinitely expand the set of axioms to try add to the things provable within it.
If you start with consistent mathematics, you are correct that it is possible to create unprovable truths within that system. If you then add God to that set of axioms, to prove those things that were unprovable truths in the original set of purely mathematical axioms, you can trivially construct statements that are not provable in that system of axioms, even though God is one of the axioms. Therefore, if you accept God as an axiom, you necessarily are accepting a lot of other ideas that are at least as unusual and inconceivable as the God axiom. So the question becomes, do we limit our set of consistent axioms purely to mathematics, whose provability is limited by the universe we live in but which are generally well-behaved in a theoretical sense, or do we expand the set of axioms to include God, and accept the torrent of strange consequences that necessarily generates, some of which may be strange or unacceptable to religious individuals who would nominally be willing to accept the idea of a God axiom in the Theory of Everything?