Can God ask a question that He Himself cannot answer?

Yes, that question. I don't understand the question, and how it differs from the question about the stone that God cannot lift. But you can search here on the site for my answer to the question about the stone. In fact, this is a weaker question than the stone. Even if he says he cannot ask such a question, there is no flaw in his ability, compared to the inability to create a stone that he cannot lift. But there is no point in agonizing over this unnecessary question.