Gödel's proof of the existence of God

Gödel did indeed produce such a proof, an argument of the type known as ontological. The proof can be found in the Collected Works, vol. III, p.403. General information about ontological arguments for the existence of God is to be found in the on-line Stanford encyclopedia of philosophy. Christopher Small has an extensive presentation and discussion of Gödel's ontological argument.

Gödel's ontological argument doesn't have anything to do with the incompleteness theorem or its proof. It shouldn't be assumed that the God referred to in the argument has much to do with God as conceived in any theistic religion. Gödel was a great admirer of Leibniz (who also had an ontological proof) and believed in the possibility of a "rational theology".