Bullshittiness is a quality of a statements by a person. The statement alone has a truth value, but its utterance by a person at a particular time is informed by the context including mostly the intention of that person (how much they care about the truth value of the statement).
If taken literally, a question is not a statement at all but a request of a statement with a truth value.
But utterances are utterances; they presume consequences and evoke feelings in people related to truth values. "Have you stopped beating your wife?" assumes that you were, which is not necessarily the case. "Are you crazy?" Isn't really asking for a response but intended to show someone that the speaker does not approve.
I think it is not common or easily ascribed, but I think yes, one can make a question that has a bullshit value. The intent of the question would have to be that the truth value of the assumptions or of the answers is lacking in relevancy. Then it would be fairly reasonable to say "Hey, that question is bullshit!".
For example, if a monk asked another "Have you stopped beating your wife?" or a psychiatrist asked a patient "Are you crazy?". Both would be examples of bullshit questions
Can you ascribe the attribute of 'bullshit' to a question?