Should he marry his cousin in secret?
Praise be to Allah.
The presence of your cousin with you in the house, when she does not wear hijab and you see her and are alone with her is something that is haraam and is not permissible. When Islam forbade women to show their adornments and take off the hijab, and made it haraam for a woman to be alone with a non-mahram man, it forbade these things so that the matter would not develop further, as appears to be the case from your question.
What you must do is fear Allah, your Lord, and take precautions to keep yourself safe and far away from sin.
Either this woman should leave and live somewhere else, but it should be close to your family so that your father will be able to check on her at any time and be reassured of her welfare.
Or you could marry her, but it should be a valid marriage with the permission of her father. With regard to the idea that her father refuses to let her get married before she has completed her studies, if you have not heard that directly from him, then you should ask her father for her hand in marriage. That will undoubtedly be better than the situation you are in at present. We ask Allah, may He be exalted, to help and guide you both.
With regard to what you are asking about, namely marrying her in secret, this is not permissible for the following reasons:
Firstly: the approval of her guardian is one of the conditions of marriage being valid, as has been explained previously in fatwa no. 7989. If you marry her without the knowledge of her guardian, this will make the marriage invalid.
Secondly: marriage must be witnessed or announced openly, but you want it to be a secret marriage that no one knows about. A secret marriage that is not witnessed or announced openly is also an invalid marriage. Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
As for getting married in secret, in which the parties agree to conceal it and not bring anyone to be witness to it, this is invalid according to most scholars, and it comes under the heading of illegal sexual intercourse. Allah, may He be exalted, says (interpretation of the meaning):
“All others are lawful, provided you seek (them in marriage) with Mahr (bridal money given by the husband to his wife at the time of marriage) from your property, desiring chastity, not committing illegal sexual intercourse”
End quote from Majmoo‘ al-Fataawa (33/158)
This has also been discussed previously in fatwa no. 111797
And Allah knows best.