Born in Maybole, Ayrshire, McQuater was most notable for his work in the United Kingdom with Bert Ambrose in the 1930s, and also for some recordings made with George Chisholm and Benny Carter.

[1] McQuater turned professional in his teens and got a regular position with Louis Freeman's Band, which played at Greens Playhouse in Glasgow.

The following year he joined Lew Stone's band[2] and made the classic recording of "Pardon Me, Pretty Baby".

He freelanced widely in the 1960s, with sessions for radio, television and films and appeared regularly with Jack Parnell's ATV Orchestra.

His TV work included his playing the featured trumpet parts in The Muppet Show, throughout its 1976 - 1981 run.