Beachgoers spotted the animals on the beach at Anastasia Island State Park on Sunday and called authorities. The mom, about 14 feet long, died on the shore.
The 4-foot-long calf was still alive when Florida Fish and Wildlife officers arrived, but was euthanized because it was not old enough to survive on its own.
Authorities will take the carcasses to Marineland in southern St. Johns county where a necropsy will be performed to try an determine why the marine mammals beached themselves.
Experts are trying to find out how the mother died.
Wildlife experts say if you spot an animal like a whale or shark on the beach, to call animal control immediately. They say a common mistake people make is to drag the animal or release it back into the ocean.