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 held to determine whether Garak was to be held responsible for his actions in the killing of four Starfleet officers. Garak was found to be not in command of his faculties as he was exposed to a psychotropic compound which enhanced his species' inherent xenophobic and paranoid tendencies and therefore not responsible for the deaths. (DS9: "Empok Nor")