Scottish centre-half McQue was in the Liverpool team that won the Lancashire league in the club's inaugural season. His claim to fame is also scoring in Liverpool's first-ever Football League match in 1893, a second division fixture away to Middlesbrough Ironopolis. He was described as "a finely built young fellow, with superb tackling powers and excellent judgement of the ball." He only missed two games as the club easily won the title by 12 points from Small Heath in second place, winning 22 and drawing six of their 28 games.

McQue initially struggled with the pace of the First Division. "Compared with last year's brilliance a great falling off is most noticeable in McQue," Liverpool Mercury reported. Liverpool weren't strong or experienced enough to cope with life in First Division, but only Tommy Bradshaw made more League appearances for the club than McQue in 1894/95. Liverpool were relegated but went straight up again as champions. McQue started the 1896/97 season as a regular but six weeks into the campaign the diminutive Robert Neill replaced him. Neill returned home the following season so McQue got more chances in his final season at Anfield, playing 19 games. McQue had an excellent eye for goal and 14 goals for a defender at that time was a terrific return.

After undergoing an operation at the Victoria Infirmary, Glasgow, McQue died only 40 years of age.