An inquest is a fact-finding investigation looking into a crime such as a murder.
In 2373, an inquest was held to determine whether Garak was to be held responsible for his actions in the murder 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")