I think it might be Grandfather Mountain, located in western NC near Blowing Rock and Boone.
This was taken from the Grandfather Mountain website:
"The original Cherokee name for the mountain was "Tanawha," meaning "a fabulous hawk or eagle." It was named "Grandfather" by pioneers who recognized the face of an old man in one of the cliffs. Many vantage points reveal different faces, so there is no one official profile of the mountain, but the most popular can be seen from the community of Foscoe, seven miles north of Linville and 10 miles south of Boone on NC 105".
Jerry.