Yes, even if up until the point where this other person had their accident with you they had a clean driving record if the accident goes on their driver's history and you make a claim the person's insurance rates could go up.

Some insurance companies may have a first accident forgiveness or only take away the person's good driver discount but because they now have an accident and claim on their record normally their insurance carrier could raise their premiums.

Whether their insurance rates would go up or not would depend upon their insurance provider's rating system. The person that hit you could find out about their rating system from their agent or possibly by contacting your state's insurance regulator since insurance companies must file their rates with them.