What if you have a fender bender accident and no police report. can the person you hit still file a claim on your insurance?

Absolutely. You have an obligation to report it to your insurance company and to cooperate with the settlement of the claim. If it was very small you can try to deal with the person directly to keep it from going on your insurance. Be carefull with this because you can't try to settle it your self then turn it over to your insurance company. You can contact them and ask them to get an estimate before you decide.