The mutants in charge of all state related excursions and who assume control in times of Conflict or War.
Captain Commander - Cyclops
Great Captain - Gorgon
Great Captain - Bishop
Great Captain - Magik
I think it goes without saying that these four are soldiers well before they're diplomats. So while the Quiet Council is clearly the Political(and judicial) section of the Government for Krakoa; the Captains, both by description and membership, make up the Military.
Now this may just be an assumption but it feels like each captain may correspond with each section of the Quiet Table. One presumably for each season (Possibly chosen by each season.)