For God to be represented by a Gödel sentence or a set of Gödel codes would require, initially, that a theory of God was required (or at least able) to prove arithmetic past a certain point. More broadly, theories of sets of such codes are theories of sharps, e.g. the most famous, and simplest, example is 0♯. But there are posits of other sharps, e.g. various Icarus sets.
Generally, some X♯ exists if and only if there exists a nontrivial elementary embedding j: L[X] → L[X]. So if God was a sharp, It would be generated modulo some n.e.e., which sounds peculiar and anyway out-of-line with the idea that God is self-existent and self-explanatory (if existent and explainable at all).
Even on the level of individual codes, I would be curious as to how, "God exists," would allow us to prove sufficiently complex arithmetical statements. Or, "God is metaphysically simple," "God is eternal," etc.: do these have an arithmetical use? Some Christian analysts do think such things (Cornelius van Til claimed that the concept of the Trinity provides a solution to the "one over many" problem, for example) but I don't know that those analysts have tried to develop such doctrines to the relevant extent (to the extent of justifying higher theories of arithmetic).