It's so very frustrating when someone doesn't believe you and you're telling the truth. I hate it! So, what can you do? This person probably has trust issues based on some experience that happened in the past. Is this fair to you? No. The more honest you are in life with all the people around you, the apter they are to believing you.
Don't let this person's opinion stick to you. Keep going regardless of what the person thinks. Don't react. Some people need more time than others to start trusting someone. Just keep on being the honest, reliable and trust-worthy person you are. In time, you will be seen as what you are.