"We ask questions about personal problems."
Translation:우리는 개인적인 문제들에 관하여 질문을 합니다.
(1) 관한 as you rightly pointed out is an qualifier-adjective, used to modify nouns only.
So: ~에 관한 질문 = relevant question on~
(2) 에 관하여/에 관해 are phrasal adverbs, thus can be used to modify nouns/verbs.
~에 관하여 질문하다(verb) = ask about ~
~에 관하여 질문을(noun) 하다 = ask questions about ~
묻다 = to ask; to enquire
질문하다 = to raise question(s); to question
• 묻다 (ask) indicates that an answer is expected, possibly from a specific entity or person
부산행 마지막 열차 시간을 물었다 I asked for the time of the last train to Busan
나는 너에 대해 물었다 I asked (enquired) about you
• 질문하다 (question) implies doubt on certain issue. The question(s) raised might not be directed at anyone in particular and does not necessarily require a reply.
사람들은 미디어의 신뢰성에 질문하다 People question the credibility of the media
"질문해도 돼요?" May I raise a question?
In colloquial usage however, the 2 expressions seem to be interchangeable.