Quinn was skills lead on the council and the main contracting voice on the industry improvement body.
His departure leaves the council dominated by consultants and clients.
A Balfour spokesperson said: “Having delivered the Skills workstream report in July and taking into account his other important long-term commitments, including The 5% Club, Leo suggested to the CLC that the moment had come to transition his role on the CLC skills workstream to new leadership.
“Balfour Beatty will of course continue its support and engagement with Government on key sector issues including the industry’s skills shortage.”
The move comes just days after Quinn launched a broadside against the CITB and confirmed Balfour would be voting against continuation of the skills body.