Is it permissible for a maternal cousin to swear on the Holy Qur’an that there is no love relationship with her paternal cousin, in order to be able to marry him, knowing that their intention is to marry?

An emotional relationship between a man and a woman who are not married to each other is not permissible, because it is a means to corruption. They must repent and end this relationship. Lying under oath is not permissible except out of a necessity that is Islamically recognized. Euphemisms can be used to avoid lying. The best solution for two people who love each other is marriage. If that is not possible, then he must turn his heart away from thinking about her, and perhaps Allah will facilitate for him someone better than her.