If my car was hit by a family member, can I claim the damage with my insurance company to help them avoid a higher rate?

Basically, only my car was damaged and it looks like it will be over $1k worth at least (quarter panel damage, door damage). I don't want my family member to worry about it, so I am okay with paying the deductible to get my car repaired through my own insurance. Is this possible? Or will they force me to file a claim against my family member? They only have liability coverage, and I don't want their rates to go up.

You could claim the damage as a hit-and-run, but that's technically not what happened so I would avoid lying to your insurance company. I would recommend getting a repair estimate first so you can factor that into your decision. The only way to avoid causing an increase to your family member's policy would be to pay for the damage out of your own pocket and avoid filing a claim altogether. Your family member is clearly at fault in the incident so ultimately their policy would have to cover the damage if a claim was filed.