What do you do if you had an accident with your mom's car and are willing to pay the damages because you don't want her insurance to be notified because she doesn't know?

You will have to negotiate directly with the other driver. Make sure you do this before they contact their own insurance company. Once they do that and give them your name and license plate, there is nothing you can do to prevent your own/your mom's company from finding out. Have the other driver get several estimates on the damage and send them to you. Negotiate for which to accept. Either pay the shop or pay the owner - you'll have to negotiate which method. The other driver may be willing to negotiate because many people believe that even no-fault accidents raise their rates.