Let's assert the existence of an omniscient being for the sake of this question. (Questioning possibility of existence is e.g. done here.)
By its very nature, the being knows it is omniscient. It also knows that it knows etc. But is there any way for it to prove its nature and therefore proving that it is not just delusional? There is no reason to prove it for us as external observers, as we know its nature by assumption. There is no reason to prove it for itself, as it already knows by its nature, and I assume "all knowing" come with "not doubting the knowledge". But given an internal observer, e.g. a observer that does not know by assumption nor by omniscience that the being is omniscient, how - if even possible - could the being prove to the internal observer that it is omniscient and not delusional.
My thoughts: I think a proof in strict logical terms is impossible. No matter how many events the being predicts correctly, the internal observer can never conclude from the finite amount of correct predictions that all (infinitely) possible predictions would be true. The internal observer may be convinced, sure, but he cannot know. Also there is no way to tell for the internal observer if the omniscient being is lying to it (a possibility I would rule out in spirit of the question, but that can nevertheless not be perceived by the internal observer), and occasionally just guessing.
In conclusion, no matter what the omniscient being does, not only can the internal observer not know if the omniscient being is omniscient or delusional in believing to be omniscient, he can also not say if the being is just randomly answering questions which all turn out to be right (the latter one may constitute a question on Philo.SE in its own right).
I should mention I have no clue about philosophy whatsoever, besides talks among friends and getting some bits here and there from Wikipedia and this site.