Yes, there is a per-message cost if you are using a Business WhatsApp API account. Facebook charges some cost + The WhatsApp API provider will charge an additional cost.
You can view the charges by Facebook here https://developers.facebook.com/docs/whatsapp/pricing/
Each API provider will charge differently, here are the charges for one of the providers, Twilio https://www.twilio.com/whatsapp/pricing/us