According to notes in [GT1] "a raffish young priest of the Church of England".
An Edward Colquitt (b. 1716) was a junior incumbent at the Cowgate Qualified Chapel in Edinburgh from 1747 to 1762.1
Also, an Edward Colquitt was the son of John Colquitt II, a tax collector in Liverpool, and Frances Smith. He was born at Leith (just north of Edinburgh) and was educated at St. John's, Cambridge, where he graduated B.A. in 1739. Also a sometime minister of St. Andrew's in Edinburgh, he died unmarried.2. He was ordained Priest by the Bishop of Lincoln 4 March 1754, and he was instituted Rector of Husbands Bosworth, co. Leicester, 12 March 1754 on the presentation of Edward Smith, and held the living until 1776. He died on May 3, 1777.3
It is unclear to the author of this site whether all these references are indeed to the same Edward Colquitt, but given the identical years of birth and the fact that the Colquitt who became Rector of Husbands Bosworth was born just outside of Edinburgh, it seems probable.
Boswell knew Colquitt from Edinburgh ca. 1760, where they were fellow members (and founders) of The Soaping Club. Boswell first mentions him in his journal of August 21, 1764 in Brunswick, when he recollected one of Colquitt's stories.