The Starfleet Inspector General Office was a specialized department within Starfleet, tasked with investigating the internal affairs of officers and the validity of their orders. The inspectors of this department were part of the command division. The Inspector General has similar responsibilities as Starfleet Internal Affairs, which investigated suspected misconduct rather than making general inspections.
The Inspector General Office had a specialized insignia utilized for uniform wear by branch admirals leading the department in the 2350s, a white shoulderboard bearing a reflective silver triangle surrounded by embroidered oak leaves, to be worn on Starfleet uniform service jackets. (FASA RPG module: Star Trek: The Next Generation Officer's Manual)
In 2364, Rear Admiral Gregory Quinn was a branch admiral in this department. In this capacity he oversaw an investigation into the behavior of Captain Jean-Luc Picard during Picard's first year in command of the USS Enterprise-D. Quinn's true motive was to determine whether the Enterprise crew, and the senior staff especially, had fallen prey to a parasitic influence that was later revealed to be bluegill infestation. Quinn utilized Lieutenant Commander Dexter Remmick as his inspector for the case. At the conclusion of the investigation, Quinn revealed his inquiries were a preliminary step towards offering Picard a promotion to admiral's rank, as a candidate for Commandant of Starfleet Academy. Picard declined the promotion. (TNG episode: "Coming of Age")
Quinn and Remmick, at some point in 2364, ran afoul of those controlled by the neural parasites. By the time the Enterprise-D returned to Earth, Quinn had been taken over by parasites, and Remmick had become host to a spawnmother. Presumably, the parasites used control of the Inspector General Office to make their victims able to commit serious breaches of policy without fear of internal command reprisal. (TNG episode: "Conspiracy")
An inspector general, Captain Farrell, was dispatched to Deep Space 9 in 2370 to make a report on conditions at that space station. While he found the station up to code, he was not impressed with the conduct and state-of-mind of the senior staff, until he found that the station crew was overtired because they had been caring for an abandoned baby. (DS9 comic: "Baby on Board")