A hearing room was a room aboard a starship where a hearing was held.
In 2267, such a room aboard the USS Enterprise was used for a hearing about Spock attempting to return his former commanding officer, Christopher Pike, to Talos IV. The room had a screen onto which footage was broadcast from the planet, showing a previous visit Spock and Pike had made there. (TOS: "The Menagerie, Part I", "The Menagerie, Part II")
The script of "The Menagerie, Part II" called for this hearing room to be a redress of the Constitution-class briefing room, which it evidently turned out to be.
In 2372, a hearing room aboard the USS Voyager was the site of a hearing to assess Quinn having requested asylum aboard that ship. (VOY: "Death Wish")