Learn from your mistake and don't do it again. You can only control how you act from here on out. Consistently be trustworthy to that person moving forward. Only they can control if they want to trust you again. You just have to be confidant that you are doing the right things now and moving forward. Be patient!
It depends on how you lost their trust and how understanding the person is. Start by apologizing and then don't violate their trust again. It may take some time. Since you don't say how you lost their trust it's difficult to give a specific answer.
Well; I'm no expert but I'll try. You don't want to bribe them. You can try to trust that person more. Then that person could see that you trust them; so they would trust you over time. I hope I helped.