Something like: to pop the question, get hitched, to exchange vows. Am I right?
The only one there that might work is "pop the question". "To get hitched" means "to get married" which they're not doing. I don't think they're exchanging vows either. I think he's simply proposing marriage. We don't know her answer so we can't say for sure that they are getting engaged. She might say no!
He is asking for her hand in marriage.
He is asking her to marry him.
He is popping the question.
He is asking if she will do him the honour of being his wife.
He is asking if he can make an honest woman of her. ;-)