The correct answer is C. count.
To count on someone means to rely on them to do something or to be there for you. In this case, the speaker is saying that they can rely on the person they are talking about to help them when they have a problem.
The other options are not as appropriate in this context. Bear means to tolerate or endure something difficult. Trust means to have confidence in someone or something. Believe means to accept something as true.
In this case, the speaker is not saying that they can tolerate or endure the problem they are having, or that they have confidence in the person they are talking about. They are saying that they can rely on the person to help them.