Conner gave up playing tennis and turned to golf, turning professional in 1971. He failed four times at
qualifying school before finally getting his tour card at
1974 PGA Tour Qualifying School. He played on the
PGA Tour from 1975 to 1989 and again in 1992. He finished second four times: in 1979 to
Hubert Green at the
First NBC New Orleans Open, in 1981 to
Dave Barr in a five-man playoff at the
Quad Cities Open, in 1982 to
Tom Watson in a playoff at the
Sea Pines Heritage, and in 1984 to
George Archer at the
Bank of Boston Classic. He won the 1988
Deposit Guaranty Golf Classic which was played opposite the
Masters Tournament - it was an official money event but not an official win on the PGA Tour. His best finish in a
major championship was a T-6 at the
1981 U.S. Open. Conner played on the PGA Tour's developmental tour from 1990–91 and 1993-95. He won twice in 1991 at the
Ben Hogan Knoxville Open and the
Ben Hogan Tulsa Open and finished fifth on the money list to regain his PGA Tour card. After turning 50 in 1996, Conner played on the
Senior PGA Tour from 1996 to 2002. His best finishes on this tour were a pair of second places: at the 1997
The Transamerica and the 1998
Kroger Senior Classic in a five-man playoff. == Awards and honors ==