If you make an insurance claim for a minor paint job repair after an unreported accident will it appear on your car's Carfax record?

Almost without fail any claim you make with your insurance carrier will show up on a C.L.U.E. report. I don't know from where Carfax gets their information but it is a good bet that it will be entered into the national database. I take it you mean that the accident was unreported to the police. That makes no difference to the C.L.U.E. database. No insurance company provides data to Carfax unless it is a total loss. Some repair shops report data - so where you get it fixed might report the paint job.