Quaid was born in Houston, Texas, to Juanita Bonniedale "Nita" (née Jordan; 1927–2019), a real estate agent, and William Rudy Quaid (1923–1987), an electrician. Quaid has English, Scots-Irish, and Cajun ancestry. Through his father, Quaid is a first cousin, twice removed, of cowboy performer Gene Autry. Randy Quaid grew up in Bellaire, Texas, a small city surrounded by Houston, and in southwest Houston. He is the older brother of actor Dennis Quaid.
In high school, he took a class in drama on a whim, although he didn't expect he would enjoy the lectures. After the third day, however, he was captivated by the course and decided to make acting his professional goal. He continued studying acting at the University of Houston. During one course, his teacher sent him to audition for Peter Bogdanovich, who was casting for The Last Picture Show, and Quaid won the role in what became his debut film.