Regarded as one of Cork's greatest-ever players, Quirke made his first appearance for the team during the 1932 championship and was a regular member of the starting fifteen until his retirement after the 1946 championship. During that time, he won four All-Ireland medals, four Munster medals and two National League medals. Quirke was an All-Ireland runner-up on one occasion.
At club level, Quirke was a three-time county club championship medalist with Blackrock.
Quirke's sister, Dolly Leahy, was an All-Ireland medalist in camogie with Cork.
Quirke played his club hurling with the famous Blackrock club and enjoyed much success.
In 1929, he was just out of the minor grade when he lined out in his first championship decider at senior level. A 5–6 to 2–2 defeat of fierce city rivals St. Finbarr's gave Quirke his first championship medal.
Blackrock reached the county final once again in 1930. A 3–8 to 1–3 defeat of Glen Rovers gave Quirke a second successive championship medal.
1931 presented Blackrock with the chance to complete a hat-trick of championship titles. A decisive 2–4 to 0–3 of Éire Óg gave Quirke his third successive championship medal. Blackrock would not win another championship for twenty-five years.
Quirke first came to prominence on the inter-county scene as a member of the Cork minor hurling team in 1929. His sole season in that grade ended with a Munster semi-final defeat by Tipperary.
Three years later, in 1932, Quirke made his senior championship debut for Cork in a Munster quarter-final defeat of Waterford. His first few years on the team coincided with a downturn in Cork's hurling fortunesl
By 1939, Quirke was one of the more senior members of the Cork team as Cork made a breakthrough in the provincial championship after nearly a decade in the doldrums. A narrow 4–3 to 3–4 defeat of Limerick gave Cork the title and gave Quirke his first Munster medal. The subsequent All-Ireland final pitted Cork against Kilkenny. In one of the most iconic championship deciders of all-time, played on the day that World War II broke out, the climax of which was played in a ferocious thunder storm. While a draw looked likely as the hour drew to a close, Paddy Phelan sent a seventy-yard free in towards the Cork goalmouth. The sliotar was gobbled up by the defence and cleared, but only as far as Jimmy Kelly who sent it straight over the bar for a one-point lead. Immediately after the puck-out, the referee blew the whistle and Cork were defeated on a score line of 2–7 to 3–3.
Although defeated in the All-Ireland decider, Cork continued their breakthrough in 1940. An 8–9 to 6–4 defeat of Tipperary in the decider gave Quirke his first National Hurling League medal. He added a second winners' medal to his collection in 1941 following a defeat of Dublin.
An outbreak of foot and mouth disease severely hampered the 1941 championship. As a result of this, Cork was nominated to represent the province in the All-Ireland series. Quirke was on the team for the final against Dublin, however, it turned into a one-sided affair thanks to contributing goals from Quirke himself and Ted O'Sullivan. At the full-time whistle, Cork had won by 5–11 to 0–6. It was one of the most one-sided championship deciders of all-time, however, it did give Quirke an All-Ireland medal.
Quirke added a second Munster medal to his collection in 1942 as Tipp were downed by 4–15 to 4–1. The All-Ireland final was a replay of the previous year with Dublin providing the opposition once again. The game was a close affair with just a point separating the sides at the three-quarter stage. In the end Cork won comfortably enough by 2–14 to 3–4 and Quirke collected a second All-Ireland medal.
A 2–13 to 3–8 defeat of Waterford in 1943 gave Quirke a third Munster medal. He later lined out in a fourth All-Ireland decider with Antrim becoming the first Ulster side to qualify for a final. Unfortunately, the occasion got to the Glensmen as Cork built up an unassailable 3–11 to 0–2 half-time lead. The final score of 5–16 to 0–4 gave Cork their second-ever hat-trick of All-Ireland titles while it also gave Quirke a third All-Ireland medal.
In 1944, Cork was attempting to capture a fourth All-Ireland title in-a-row. No team in the history of the hurling championship had won more than three consecutive titles. The year got off to a good start when Cork defeated Limerick by 4–6 to 3–6 after a replay to give Quirke a fourth Munster medal. For the third time in four years Cork faced Dublin in an All-Ireland decider. Joe Kelly was the hero of the day and he contributed greatly to Cork's 2–13 to 1–2 victory. It was a remarkable fourth successive All-Ireland medal for Quirke.
Five-in-a-row proved to be a bridge too far for Cork, however, the team returned in 1946. A 3–9 to 1–6 defeat of Waterford allowed Cork to advance to another Munster final. Unfortunately for Quirke, that semi-final victory was his last championship game for Cork.
Murphy also had the honour of being selected for Munster in the inter-provincial series of games. He made his debut with the province in 1936 and was a regular until his retirement in 1945.
In 1936, Quirke was one of only two Corkonians to line out in the inter-provincial decider. A narrow 2–8 to 3–4 defeat by fierce rivals Leinster was the result on that occasion.
In spite of a defeat in his debut season, the province went on to dominate the rest of the decade. Defeats of Leinster in 1937 and 1938 gave Quirke his first two Railway Cup medals.
Quirke was dropped from the starting fifteen in 1939, however, he won a third Railway Cup medal as a non-playing substitute following a 4–4 to 1–6 defeat of Leinster. He was reinstated to the starting fifteen the following year and won a fourth winners' medal as Leinster were downed once again.
Munster surrendered their title to Leinster the following year, however, the province went on to dominate the rest of the decade once again. Defeats of Leinster (1942, 1943), Connacht (1944) and Ulster (1945) brought Quirke's Railway Cup medal tally to eight.Blackrock
Cork Senior Club Hurling Championship (3): 1929, 1930, 1931
All-Ireland Senior Hurling Championship (4): 1941, 1942, 1943, 1944
Munster Senior Hurling Championship (4): 1939, 1942, 1943, 1944
National Hurling League (2): 1939–40, 1940–41
Railway Cup (8): 1937, 1938, 1939 (sub), 1940, 1942, 1943, 1944, 1945 (sub)