I don't think so.
It looks (from the Oxford Latin Dictionary) as though rogāre means to ask someone a question; to ask someone for his opinion; to request things; but those seem to be distinct from the (very interesting) "questioning OF the gods," i.e., calling them (their existence, I think you mean?) into question.
(Unless I'm simply missing something in that entry.)
Yes, calling them/their existence into question is one possible meaning of the sentence ("Don't question the gods") I am asking about! Rather than their existence, though, I was thinking more along the lines of not questioning/doubting their decisions/commands/actions, much in the way one might be told not to question ones parents or an elder/superior/authority figure.
noli -> "don't" when directed at one person. It's the second person present imperative singular. "Marcus, don't ask the gods!" -> Marce, noli deos rogare!
nolite -> "don't" when directed at multiple people. It's the second person present imperative plural. "Marcus and Livia, don't ask the gods" -> Marce et Livia, nolite deos rogare!