Ruling on teaching Qur’aan to a non-Muslim
Question

My question is i have a close friend, who is intending to convert after she finishes her studies, she is very interested in Islam, and I've been teaching her verses such as Al-fatiha and Ayat Al-Kursi off by heart, and she has very willingly learnt them off by heart, is this permissable though because at the moment she is not muslim yet?
Praise be to Allah.Firstly: 
There is nothing wrong with teaching Qur’aan to a non-Muslim if there is the hope that he will become Muslim and benefit from this teaching. 
Al-Nawawi (may Allah have mercy on him) said: 
Our companions said: The kaafir should not be prevented from listening to the Qur’aan but he should be prevented from touching the Mushaf. But is it permissible to teach him Qur’aan? That depends. If there is no hope that he will become Muslim, it is not permissible, but if there is the hope that he will become Muslim, it is permitted according to the sounder opinion; this was the definitive view of al-Qaadi Husayn, and was regarded as more correct by al-Baghawi and others. End quote. 
Al-Majmoo‘, 2/85 
Ibn Hajar (may Allah have mercy on him) said: 
This is an issue concerning which the early generation differed. Maalik did not allow teaching Qur’aan to a kaafir; Abu Haneefah gave a concession allowing it; and the view of al-Shaafa‘i differed. What seems to be the correct view is that there should be a differentiation between the one who it is hoped will become Muslim and there is certainty that he is not learning it in order to criticize it, and the one who it is certain it will have no positive impact on him or it is thought likely that this will enable him to attack Islam. And Allah knows best. A differentiation should also be made between teaching a little of it or a lot. End quote. 
Fath al-Baari, 6/107 
Allah, may He be exalted, says (interpretation of the meaning):
“And if anyone of the Mushrikoon (polytheists, idolaters, pagans, disbelievers in the Oneness of Allaah) seeks your protection then grant him protection so that he may hear the Word of Allaah (the Qur’aan) and then escort him to where he can be secure, that is because they are men who know not”
[al-Tawbah 9:6]. 
Al-Shawkaani (may Allah have mercy on him) said: 
What is meant is that if any of the mushrikeen whom you are commanded to fight seeks your protection, then give him protection, i.e., protect him and keep him safe so that he can hear the Word of Allah from you and ponder it properly and understand the truth to which you are calling him. Then escort him to a place of safety, i.e., a place where he will be safe after hearing the word of Allah if he does not become Muslim. End quote. 
Fath al-Qadeer, 2/491. 
Secondly: 
Islam does not approve of a relationship and meetings between a young man and a girl that are outside the framework of marriage. So you have to repent to Allah from what you are doing. If you hope that this girl will become Muslim, then our advice to you is that you should marry her so that you can help her to become Muslim, and a reward like hers will also be written for you. 
We ask Allah to guide and help you. 
And Allah knows best.