In the year 2364 Admiral Quinn was attached to Starfleet Operational Support Services and made, among others, the executive decision to postpone the planned upgrade of the warp sensor grids of the USS Gremlin due to budgetary considerations. (TNG: "Conspiracy")
Most importantly, Quinn played a crucial role in the uncovering of an attempt to take over Starfleet Command by parasitic beings in that year. Quinn ordered the Inspector General officer Dexter Remmick to investigate whether or not the USS Enterprise-D had been infiltrated also. (TNG: "Coming of Age")
Nothing was discovered at the time, but Quinn himself would later become a victim of the alien parasite infestation. His mind was overtaken by the hitherto unknown parasitic intelligence when it infiltrated Starfleet Headquarters. Later that year, Captain Jean-Luc Picard and Commander William T. Riker of the Enterprise-D destroyed the alien mother creature which lived in the body of Dexter Remmick, thereby also causing the deaths of all other associated parasites on Earth. Quinn himself was expected to make a full recovery following the ordeal. (TNG: "Conspiracy")
Background information Edit
Quinn was never seen or mentioned again after the first season of The Next Generation, with the exception of archive footage used in TNG: "Shades of Gray". Quinn's recovery is stated in a log entry made by Captain Picard.
Quinn returned, one rank higher, as a full rear admiral in issues of DC Comics' TNG series story arc entitled War and Madness.