"What do you do when you have a problem with a co-worker?"
Usually when two co-workers have a problem, then they will just hate each other and try to be better than the other. You can't say this because it is the wrong way to handle it. I would probably not deal with the problem and hate the other person, but that is not what I would say in an interview. You can mention that you believe having to deal with a problem with a co-worker is difficult because competitiveness comes into factor. Along with this statement, follow up with how you would try to look beyond that to make your relationship better.
Short Answers
"If I have a problem with a co-worker, I believe it's so important to talk to them. Even if I'm the one with the problem, I want to know so I can be a better team mate."
"If I have a problem, I try to approach my co-worker and let them know as professionally as possible. If I'm the one with the problem, I'll politely ask what I can do to improve the relationship. I found that proper communication solves many problems."
"I've always been told that I'm a great person to work with, but if I were in a situation where I was having a problem, I believe I would work on our relationship by being more understanding."
Long Answer
"Well it all depends on if it's a personality conflict or an issue that is causing the problem. An issue is easy to deal with. I think I'd go and talk with the co-worker saying something like, 'If you have some time, I think it will be good for us if we talk about this issue.' Starting a conversation to deal with the problem with a co-worker is the hardest step, and if it's done properly, then the issue can easily be resolved. If it is a personality issue, then I would try to be more understanding and at worst deal with the problem by anticipating conflicts ahead of time to put myself in a position to either avoid it or to make it less of a problem."
This is showing answers by using scenarios. It is a solid answer but if you disagree, feel free to think of a good scenario that will fit you better.