Is there a prohibition against reading the Qur’an approximately five or ten minutes before the Adhan of the Maghrib prayer?

There is no reported prohibition against reciting the Quran at the time mentioned. Rather, there are commands to engage in remembrances (dhikr) during it, which often include Quranic verses. Generally, reciting the Quran is permissible at all times and in all situations, unless the person is in a state of ritual impurity (janabah), or in a filthy place, or has an impure mouth before rinsing it, or during bowing (ruku') and prostration (sujud), as recitation is prohibited in these two postures. It has been authentically narrated from Ali (may Allah be pleased with him) that nothing prevented the Messenger of Allah (peace and blessings be upon him) from reciting the Quran except ritual impurity (janabah).