質問 refers more to an actual question. 問題 is more like "problem", and while it can be used in the context of being a question, it can also be used in other situations. In anime it's common to hear things like "問題があります！" ("There's a problem!"), as an example.
Also, 問題 is frequently use in combinaison with other words/kanjis to talk about debate, research, social problems, etc. In other words, anything that is more than just a simple question. If you type "*問題*" with the two *, you will find a ton of words like that. エネルギー問題 -> Energy problems/issues 社会問題 -> Social problem
And I don't think that even half of them are on renshuu since their are just combination of words/kanjis that don't need a definition. Once you got the pattern tho, they're easy to recognize, so don't worry about that.
