Joe Quaid was born in Ahawilk Feohanagh, County Limerick in 1972. He was born into a family that had a strong association with hurling. His father, Jim Quaid,was a key player on the Limerick team which beat London in the 1954 junior All-Ireland final. His uncle, Jack Quaid, was also a member of this side and the brothers also won Munster Senior Hurling medals in 1955, when a young Limerick side surprised a more experienced Clare side in the final. Their first cousin, Seamus Quaid, also played on Limerick senior teams and also played with Wexford, with whom he won an All Ireland senior medal in 1960. Joe succeeded his cousin playing position, Tommy Quaid, who was the goalkeeper on the Limerick team from 1976 until 1993.
In the late 1980s Quaid joined the Limerick minor hurling team. He later moved on to the inter-county under-21 side, however, he had little success at either level. In 1991 Quaid joined the Limerick senior hurling team as understudy to his cousin Tommy Quaid. He took over as the first-choice goalkeeper in 1994, the same year that he won his first Munster title. Limerick later qualified to meet Ofaly in the All-Ireland final. The game has gone down in history as "the five-minute final." Limerick were leading by five points with five minutes to go when Offaly's Johnny Dooley scored a goal which seemed to paralyse the Limerick team. Another 1-5 followed and victory was effectively snatched from Limerick. As the goalkeeper who let in two late goals Quaid was blamed by many Limerick people for costing the county an All-Ireland title. In spite of this Quaid was still honoured with an All-Star award, a decision which may have been influenced by his splendid displays earlier in the campaign, especially in the game against Cork when he made many stunning saves.
Two years later in 1996 Quaid captured his second Munster medal and once again his team qualified for the All-Ireland final. This time the opposition was Wexford. Once again Quaid ended up on the losing side and once again he had to shoulder most of the blame for costing Limerick another All-Ireland title, though some would not agree with this statement as he made some brilliant saves in that game. Possibly because of this he was presented with a second All-Star award. The following years proved difficult as Limerick failed to recapture any more Munster titles and Quaid battled injury. In a National Hurling League game against Laois in 1997 Quaid suffered a terrible injury when the ball struck Quaid in the testicle at high speed following a penalty attempt by Paul Flynn Waterford. His right testicle exploded on impact while half of the other testicle had to be removed. The injury could have cost Quaid his career, however, he battled back and continued hurling. Quaid was quoted after the incident to have said "Ouch".
Following the conclusion of the 2000 championship Quaid retired from inter-county hurling. He returned in 2002 for one season but he had no more success. Quaid won an All Ireland Hurling All Star in 1994 and again in 1996.
Like both his cousin Tommy Quaid and his Uncle Jack Quaid before him, Joe was also a member of the Munster Team in 1995 and 1997. In 1995 he won a Railway Cup medal as Munster's goalkeeper. Munster defeated Leinster 2-25 to 5-12 in the semi final of that year, before going on to beat Ulster 0-13 to 1-09 in the final.