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.
First of all, your wife should fulfil her obligations towards you as her husband and do her house duties and this is better for her than travelling even if this is for the purpose of memorizing the Quran.
There is no doubt that the intention of memorizing the Quran is a good intention for which a person is rewarded if he does so for the sake of Allaah. However, if this leads to neglecting what is more important, especially the religious obligations, then the travelling in this case is not recommended. The Prophet said: "A woman who performs her five prescribed daily prayers, fasts the month of Ramadan, guards her chastity and obeys her husband, will be addressed (on the Day of Resurrection): Enter Paradise from any gate you like." [Ahmad]
If a person has a sincere intention of memorizing the Quran, he may do so in his country where he lives, and even in his house, especially that the means of education and learning are so many and various, and are easy to access.
Furthermore, there are two conditions which should be met for a woman to travel abroad in order to seek knowledge:
1- She must be accompanied by a Mahram (i.e. a spouse or a person whom one is permanently prohibited from marrying due to suckling, marital, or blood relations).
2- She must have the permission of her husband.
For more benefit on the ruling of a woman who travels to study, please refer to Fatwa 81210, and for more benefit on the travel of a woman in general, please refer to Fatwa 87480.
Finally, it should be noted that even if these two conditions are met, it is more than likely that leaving your home, your husband and children (if any) will cause the negligence of some of your duties and obligations. Therefore, we advise you to endeavour to memorize the Quran and your husband should help you in this regard, and you should do so in the country where you live.
Allaah Knows best.