The question is grammatical but I can't think of a context in which I would use it.
What time does the quiz start? (This would work at a pub, for example, while waiting for the pub quiz to start.)
What time is the quiz [show] on? (This would work if you were talking about a TV show.)
When is the quiz? (This would work as a general question to which the answer would be something like "Thursday evenings".)