If the cause of the fender bender was your fault then if you make an insurance claim or the person you hit made an insurance claim on the small fender bender then yes in most cases your auto insurance rates will increase for at least a little while.
Sometimes it's best to just pay direct out of pocket to the person you hit for the damages to their vehicle and then leave the auto insurance company out of it.
However usually the auto insurance will not go up right away after a small fender bender and sometimes the insurance rates will stay the same after small fender bender happens.
If the other person is at fault and you make an insurance claim against them then you insurance rates should not increase.
But the other persons auto insurance might increase at least a little bit.