Answer:
No, a question can not be answered with a question.
Related Information:
To answer a question with a question is, not only ill mannered and rude, but violates the purpose of language, which is to communicate.
A question obligates (calls for) the person who is questioned to answer. For reasons of politeness and for reasons of communication, to answer a question with a question fulfills neither the obligation of politeness or of communication.
To answer with a question is to give no answer at all. The result is that the questioner has to deal with a vague subjective sentence that may or may not have any relevence to the question that was asked.
It is simply a rude ploy to avoid responsibility for communicating effectively.
Using a well thought out statement to respond to a question takes advantage of an opportunity to act responsibly and to satify the needs of your questioner.