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.

From 《新汉英大辞典》 ------------------- 质问 [zhì wèn] 1. question; interrogate; query; call to account: bring sb. to account; put a query to...; / 提出质问 hurl a question at sb.; / 质问某人 ------------------- 问题 [wèn tí] 1. （需回答的题目） question; problem: solve a problem; / 解答问题 ask a question / 提问题 2. （需研究解决的矛盾等） problem; matter: the heart of the matter; key to the question; key to the solution of a question; / 问题的关键 essence of the problem [question]; / 问题的实质 3. （事故或意外） trouble; mishap; sth. wrong: troubleshooter; / 解决麻烦问题的能手 without any mishap; / 没出什么问题

质问，commonly use as Verb, means question(v.)or query sb. according to the fact. It's a literary word,not very often used in oral. 例句：王老师质问小明：“你为什么打他？” Eg.Teacher Wang questioned Xiaoming "why did you beat him?” 问题，commonly used as noun,means question,issue,ploblem. Could used both in writing and oral. 例句：这是个很简单的问题。 Eg. This is an easy quesiton. Hope could be helpful.

质问 means a verb. 问题 means a noun.

质问 v = interrogate, inquire 问题 n = question, problem They're not the same part of speech.