Conley attended Washington State University, where (as he told The Boston Globe in 2004) students “kidnapped” him during a recruiting visit in an effort to convince him to matriculate. In 1950 he played on the Cougar team that reached the College World Series. In basketball, Conley was twice selected honorable mention to the All-America team, leading the team in scoring with 20 points per game. He was a first-team All-PCC selection in 1950.

During the summer, Conley pitched semiprofessional baseball in Walla Walla, Washington, in which scouts from almost every Major League Baseball team came to recruit him. He also was getting contract offers to play professional basketball from the Minneapolis Lakers and the Tri-Cities Blackhawks. At first he declined the offers, saying that his family didn’t want him to sign any professional contracts until he finished school. But the offers were getting bigger, and in August 1950 he signed a professional contract with the Boston Braves for a $3,000 bonus.

Conley attended spring training in 1951 and was assigned to Hartford of the Eastern League by the request of former Braves star Tommy Holmes, who was managing the club. After a month, Conley had a record of five wins and only one loss and was praised by observers in the league, saying that he had the best fastball since former pitcher Van Lingle Mungo played in the league in 1933. On June 10, he threw a one-hitter against Schenectady Blue Jays, giving up the lone hit in the seventh inning. Holmes was promoted to manager of the Braves on June 25, and was replaced by future Baseball Hall of Famer Travis Jackson.

On April 26, 1952, the Boston Celtics selected Conley with the 90th pick of the NBA draft. Playing 39 games as a rookie in the 1952-53 NBA season, Conley averaged about 12 minutes a game for a Celtics team that went 45–26 in the regular season under Red Auerbach. Conley did not play in the Celtics’ two playoff series that season, with the team losing 3–1 in the Eastern Division finals to the New York Knicks.

Conley would return to the majors in 1954 with the Milwaukee Braves, going 14–9 in 28 games with a 2.82 ERA, making the National League All-Star team and finishing third in Rookie of the Year voting behind Wally Moon and Ernie Banks, with Conley’s Braves teammate Hank Aaron finishing fourth.

The following season in 1955, Conley would be named to the All-Star game again, completing the season with an 11–7 record with a 4.16 ERA. Conley would pitch for the Braves through 1959, compiling a record of 42-43 including an 0–6 record in his final season in Milwaukee.

Conley was the winning pitcher in the 1955 All-Star Game and was selected for the 1954 and 1959 games.

In his lone postseason appearance in the 1957 World Series on Oct. 5 against the New York Yankees, Conley pitched an inning and two-thirds in relief of starter Bob Buhl, surrendering a two-run home run to Mickey Mantle as the Yankees went on to win the game 12–3; but with the Braves winning the series in seven games.

After new contract talks bogged down, on Dec. 15, 1960 the Phillies traded Conley to the Red Sox; when he debuted with the Red Sox on April 28 against the Washington Senators, Conley became the first athlete to play for three professional teams in the same city along with the Celtics and his short stint with the Boston Braves in 1952. In three seasons with the Red Sox through 1963, Conley had a 29–32 record, with the win total including the final start of his major league career on Sept. 21, 1963, going six innings against the Minnesota Twins in an 11–2 victory.

When Abe Saperstein’s American Basketball League (1961–62) was born in 1961, Tuck Tape Company owner Paul Cohen purchased a franchise, gave it the Tapers name, and placed it in Washington, D.C.; the team played its games in the Washington Coliseum. Conley signed with the team. While with the Tapers, Conley often accompanied Cohen on sales calls for his company and gained industry experience.

The Washington Sports Hall of Fame included Conley in its 1979 class of inductees.

In the spring of 1951, Conley married Kathryn Dizney whom he met the previous fall. They had three children and seven grandchildren. In 2004, his wife released a biography of Conley called One of a Kind that chronicled his life in both baseball and basketball and related how his family dealt with his being gone for most of the year.

In the days following July 27, 1962, Conley made headlines after exiting a Red Sox team bus that was stuck in New York City traffic with teammate Pumpsie Green to find a restroom, with the bus driver subsequently driving away without the players on board. As Conley recollected the episode in a 2004 interview with the Boston Globe: “So we got off and went in this bar, and when we came back out, Pumpsie said, ‘Hey, that bus is gone,’ and I said, ‘We are, too!'” Conley and Green checked into a hotel, with Green rejoining the team the next day in Washington, D.C., but Conley taking a hiatus during which he attracted media attention in attempting to fly to Jerusalem. As told by Conley, Red Sox owner Tom Yawkey fined him $1,500 with the promise he would refund the money at the end of the season if Conley rededicated himself to the team, with Yawkey fulfilling the promise in September.

“When I look back, I don’t know how I did it, I really don’t”, Conley was quoted saying in 2008 by the Los Angeles Times, on playing two professional sports in tandem. “I think I was having so much fun that it kept me going. I can’t remember a teammate I didn’t enjoy.”

Until December 2009, Conley lived in Clermont, Florida, where he played golf and watched the Orlando Magic play in his free time. He moved to his vacation home in Waterville Valley, New Hampshire, in 2010.

Conley died of congestive heart failure at his home in Foxborough, Massachusetts on July 4, 2017.