Place rib portions into the bottom of a slow cooker. Cover with water. Add carrot, celery, onion, and bay leaf. Cover and cook until tender and no longer pink in the centers, 8 hours on Low or 4 hours on High.
Meanwhile, preheat an outdoor grill for low heat and lightly oil the grate.
Take the cooked ribs out of the slow cooker and brush generously with the BBQ sauce.
Cook ribs on the preheated grill just until nicely browned, 15 to 20 minutes.