Graham was born Bette Clair McMurray, in Dallas, Texas, to Jesse McMurray, an automotive supply company manager, and Christine Duval.[2] She was raised in San Antonio and graduated from Alamo Heights High School.[3] She married Warren Audrey Nesmith (1919–1984) before he left to fight in World War II. While he was overseas she had a child (Robert Michael Nesmith, born December 30, 1942). After Warren Nesmith returned home, they divorced (1946).