Quest is a native of Liverpool, England, having been born and partly raised there. He was educated at the state comprehensive Roundhay School in Leeds, followed by Airedale and Wharfedale College and University of Leeds, where he studied law, taking his degree in 1985 (Law LLB Hons.), and was called to the Bar. He had already gained broadcast experience when he spent the 1983—84 academic year at Vanderbilt University in Nashville, Tennessee. As news director for campus radio station WRVU, he built the reporting staff from scratch.

Quest then became a trainee journalist at the BBC in 1985, joining their financial section in 1987, and moving to New York City in 1989 to become their North America business correspondent.

Quest later worked for the BBC from the USA as part of their then fledgling BBC News 24 channel. He was their business correspondent reporting on, and discussing the world stock market in a regular segment entitled World Business Report usually aired between 2:00 and 3:00 a.m. (GMT), a programme that he presented alongside Paddy O'Connell. He was also an occasional presenter on the BBC's early morning Business Breakfast programme.

Quest joined CNN in 2001 for the launch of Business International. Since this time Quest has covered a variety of different events for CNN, amongst others an analysis of the U.S. elections as American Quest and the start of the circulation of euro banknotes and coins on 1 January 2002 and the last official commercial flight of Concorde. He has also headed up CNN’s coverage of several events involving the British Royal Family.

In addition to anchoring the five-times-weekly business program Quest Means Business from London, Quest hosts a monthly program on CNN International called Business Traveller. Quest formerly anchored the monthly show Quest and the daily show CNN Today which is geared towards morning audiences in Europe. Max Foster took his place on CNN Today.

On 18 April 2008, Quest was arrested at 3:40am in Central Park, New York City, and charged with loitering and criminal possession of a controlled substance: crystal methamphetamine. He subsequently entered a drug rehabilitation facility.

Quest identifies himself as gay and is Jewish.

His twin sister, Caroline, is Managing Director of Innovation and Enterprise, at Queen Mary, University of London.