Mediate, a five-time winner on the tour, made eight birdies in his bogey-free round.
"You gotta be aggressive this week because it's going to take a low score to win," Mediate said. "I hit driver on every tee, no matter what."
Austin, the 1995 Buick Open winner, won the St. Jude Classic in Memphis three weeks ago.
Quigley is aiming for his first victory since he joined the tour in 1997. The highlight of his round was an eagle at the par-four 14th hole.
A bogey at No. 5 prevented Quigley from taking a share of the lead.
"You see the scores low every year," Quigley said. "I think you get used to seeing that (at this tournament)."