An inquest was a fact-finding inquiry into a suspicious death, such as a suspected murder, or other major incident where there is a public interest.
In 2373, an inquest was to be held to determine whether Garak was to be held responsible for his actions in the killing of four Starfleet officers.
It was expected that he would be found to have not been in command of his faculties at the time of the incident, as he had been exposed to a psychotropic compound which had enhanced his species' inherent xenophobic and paranoid tendencies, and would therefore not be held responsible for the deaths.
When Miles O'Brien told Garak that an inquest would be forthcoming, Garak replied that he'd been informed there would be already. O'Brien was confident that Garak would be cleared once the facts about the incident came out. (DS9: "Empok Nor")
Although this incident was not mentioned again, Garak appears in subsequent episodes, suggesting he was cleared.