If the tree just falls down during the Hurricane but does no damage to vehicles or your home or garage then the homeowners insurance will usually not cover the cost to have the tree cut up and removed and that usually has to come out of your own pocket.
If the tree does fall and damages a garage, home or vehicle then yes the homeowners insurance will usually cover the cost to have a professional come out and remove the tree from the damaged structure or vehicle.
If possible get trees removed away from your home or garage or vehicle before it can fall and do damage to the home, car or garage and possibly cause death should it fall through the home into a bedroom.