He revealed his decision to colleagues in a letter today.
Quinn's decision comes after interim CEO Gary Owens confirmed he would not be seeking appointment to the permanent chief executive officer role.
The former Republic of Ireland international took up the deputy interim CEO role in January. Quinn said he will "remain available" until a new CEO has been appointed.
Owens' decision came as a bit of a surprise as he was considered as a favourite to take the CEO role on a permanent basis.
Current Swim Ireland CEO and Olympic Federation of Ireland chairperson Sarah Keane is now seen as the front-runner for the position.
Quinn told FAI.ie: “I have really enjoyed my time working with the talented and dedicated staff in Abbottstown and I now have a better understanding of just how much great work they do for Irish football, often unheralded and seemingly always amidst a cloud of job uncertainty.
“When I joined the Association with Gary back at the end of January, we wanted to protect jobs and help implement real structural change within the FAI for the betterment of the Irish game. With the successful vote for change at last Monday’s EGM and the completed restructuring of day to day roles in HQ, the game can now look forward to a more productive future.
“Even though these changes were met with resistance by a small minority of constituents, the way is now clear for a new incoming CEO to drive football forward again. I will remain close to Irish football and I wish all of my colleagues the very best of luck in the coming weeks and months as more important changes are implemented to make our game and our association strong again. I will remain available to the FAI, as required, until the new CEO is in situ.”
