If you had hit the deer then the resulting damage would normally go through your comprehensive coverage. However since you turned to miss the deer and damaged your vehicle then it will likely be your collision coverage that will be used to cover your vehicle.

Your agent or claims adjuster should be able to tell you which coverage should be used and the deductible amount that will be due as well as how to proceed with a claim after hearing your description of the accident which includes what your car may have collided with in order to be damaged.

Whether this type of single car accident and claim will cause insurance points or a surcharge to be added to your policy will depend upon state laws as well as your own insurance carrier's rating system. Your agent should be able to inform you when you make a claim if according to your insurer's rating system your premiums could be affected.