I can't think of a time that I've been in that situation, but it's a great question.
I think I would try to work out whether I'd done anything that led them to believe what they do. I'm a firm believer that conflict often has 2 sides. If I could absolutely, hand on heart say that I had no part in making them feel that way, I would definitely not apologise. If I had really not done anything wrong and they wouldn't believe me, I would feel they just don't trust me, and that doesn't meet my expectations of my friends. So I would be having a conversation with them about my expectations of trust in a friendship. It's a 2 way street.