All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Reciting the Quran is one of the best acts by which one gets closer to Allaah, and reciting it is required at all times. It is not forbidden to recite it between 'Asr and Maghrib as there is no evidence which prohibits this; so in principle it is permissible. However, it is forbidden to perform supererogatory prayers between these two times without a reason, as clarified in Fatwa 84280.
Allaah Knows best.