Quick answer
Medical benefits and a liability claim are separate. Policy terms, your role, and jurisdiction determine the route.
Do now
Obtain policy declarations and claim numbers. Tell providers accurately that treatment relates to a crash. Ask the appropriate insurer about notice, forms, medical submissions, and wage-loss documentation. In NY, investigate the generally applicable 30-day written no-fault notice immediately.
Watch for
Do not assume the at-fault insurer will pay every bill as it arrives or that health insurance removes all auto-benefit requirements.
Official sources
NJDOBI — Auto Insurance Consumer Information, NY DFS — No-Fault FAQs, DC DISB — Consumer Auto Insurance Guide.