You can do multiple things, I would simply speak to your parents. If you are of a culture where this is not allowed, then I would elope (aka run away with him). Other than that, you could just buckle down and marry your cousin to retain your family's love and honor rather than run away with a man your parents do not approve or know about.
It's all really up to you, but from what I know of your culture, it is highly dishonorable to disobey your parents. So, you might be better off discussing how you have a boyfriend who is converting to Islam for you, and that you believe this could be a better choice than your cousin. Bring it up as a discussion, not an absolute.