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.
There is no doubt that memorizing the Quran is one of the greatest acts of worship by which one seeks to get nearer to Allaah and for which one is greatly rewarded. So if going out to the mosque does not cause any harm to you or to your husband, or render you neglectful about doing your housework, then you have to endeavour to convince your husband with all permissible means. If he allows you to go to the mosque, all perfect praise be to Allaah, however, if he insists on preventing you from going then you have to obey him as it is an obligation for you to obey him but memorizing the Quran is not an obligation. It is true that the Prophet said: "Do not prevent women from going to the mosque." [Al-Bukhari and Muslim]; but Imaam An-Nawawi in his comment regarding the above Prophetic narration said that this is not a command to men because a woman is obliged to remain in her house so she should not leave her home for any virtuous act (without his permission).
Nonetheless, you can endeavour to memorize the Quran from home with the help of some sisters who memorize it through the phone or by any other possible (and permissible) means.
Allaah Knows best.