On the other hand, Dumbledore and McGonagall both know some reason that Quirrell's nature is absolutely not to be inquired about by anyone. Dumbledore even evaded the Aurors' question on the subject. If the secret that they are guarding is that Q=V, then either they're all on V's side or they've been jinxed or blackmailed in some way. I don't find any of those possibilities credible. That implies that there is some other secret in play about Quirrell's nature. I don't think there's room for two such secrets, the other one being Q=V.

More likely, Q contains a fragment of V but Q remains in control. Q's lapses into zombieness are a side effect of what it takes to stay in control. Inquiries are dangerous because of the possibility, as with Harry under the Sorting Hat, of awakening the fragment to self-awareness. Q is valuable to Dumbledore and the forces of light because of the insight he can provide into Voldemort's history and how Voldemort thinks. Perhaps control over that piece of V will also be useful for magical reasons.