質問 and 問題: a question you ask and a problem you address
Distinguish requests for information, test items, and problems to solve through context.
A request for information
質問 is read しつもん and means asking someone for information or the question asked. 「先生に質問しました」 means that the speaker asked the teacher a question.
「質問があります」 says that you have a question. It can request an opportunity to ask, rather than warn that a problem has occurred.
A task to solve or a difficulty
問題 is read もんだい and refers to a test item or an issue requiring a solution. 「この問題を解いてください」 asks someone to solve this problem.
「支払いに問題があります」 says that there is a problem with payment. To say that you have a question about payment, use 「支払いについて質問があります」.
Asking about a problem
「この問題について質問があります」 means that you have a question about this problem. It distinguishes the item to be solved from what you want to ask about it.
The English word 'question' does not always correspond to 質問. A test's third question can be 「三番の問題」, so choose based on the setting and function.
Does 「問題ありません」 mean that there are no questions?
Show the answer and why
It normally means 'There is no problem' or 'That is fine.' To say that you have no questions to ask, use 「質問はありません」.