In my understanding, 质问 is more like you want to accuse someone or blame someone for doing something, so you ask that person a question. For example: 他质问她为什么要说谎。 (as a verb) or 面对他的质问，她只好承认自己撒谎了 (as a noun) When you ”质问“ someone, your tone can be more angry, firm or upset than just asking a general question.