A "court-martial" refers to a judicial proceeding within a military or quasi-military organization, or the officer(s) who sit as a court to conduct the proceeding. When a member of StarCon is accused of breaching law or regulations, a court-martial is typically convened. The court-martial reviews the evidence and determines the accused's guilt or innocence. A President of the Court oversees most courts-martial. One famous individual in StarCon who was tried by court-martial was Roger Wilco, after the destruction of the SCS Eureka and murder of Raems T. Quirk.
The court-martial scene from the introduction of SQ6 is a spoof on the court-martial in Star Trek IV, where Admiral James T. Kirk, though he had committed grievous acts against the Federation, was only demoted back to where he was most comfortable.
However, the general word choice and gravelly voice, plus the spotlight shining on Roger, were references to the Klingon show trial in Star Trek VI.