Most consumers do not fully understand the relationship between all of the various insurance companies that may be involved after your car accident.
You should always send your accident related medical bills to your health insurance company. That company should assure that the bills will be paid promptly. You do not want your medical bills going to collection.
You may also have "medical payments" coverage through your own automobile insurance company or through the car insurance of the car you were in (if it was not your car). Your bills should be submitted under this coverage as well.
Finally, the car insurance company for the person who caused the accident is also responsible for your bills. However, that claim may take months or even years to resolve.
You do not want to have your own healthcare providers go unpaid or have their payments unduly delayed.