'I'm really sorry I forgot your birthday yesterday. I'll make it up to you, I promise. Maybe we could go out for a nice meal tomorrow.'
'David and Lucy had a big argument yesterday. He bought her a big bunch of flowers to make it up to her.These sentences suggest that David was to blame for the argument.
We can sometimes say 'make up with' when one wants to repair a previously damaged relationship. 'He bought her a bunch of flowers to make up with her. The last sentence doesn't say who was to blame for the bad feelings. It just means that "He" wants to end the bad feelings/relationship.