A Q-ship is a Federation Starfleet vessel, but in disguise. From the outside, it looked like a slow-moving civilian ship. But on the inside, it had Starfleet's best engines and shields.
Calypso[]
A Starfleet Q-ship, the commercial astrogation vessel ''Calypso'' was assigned to James T. Kirk in 2380 by Admiral Kathryn Janeway to undertake a mission to identify the killers of Ambassador Spock on Primedian, Romulus, on stardate 57465.6.
Kirk's Q-ship was little more than a blunt-nosed, cylindrical main module about the same size as a single nacelle from an old Ambassador-class ship, with a slight, tapered bulge at the rear of the ventral hull, and two swept-back, outboard warp nacelles, also cylindrical, suggesting technology that was decades removed from state-of-the-art.
The bridge of this Q-ship was boxlike, rectangular, more like an enlarged interior of an old-style shuttlecraft. On the main level, four steps down from the elevated deck on which the turbolift opened, along each side of the bridge were three duty stations, each with two chairs, and two sets of displays and control boards cantilevered out from the bulkheads. But instead of the displays being aligned flat to the bulkheads, so that command staff in the center of the bridge could see each duty station at a glance, all controls faced ahead, like desks in a classroom. And what they faced were three viewscreens, side by side on the sharply inward-angled forward wall.
On the outside, it looked like a sixty-year-old derelict ship, with mismatched hullplates, three non-functional sensor grids , and an impulse drive that was out-of-balance. But on the inside, it was equipped with warp core of a Defiant-class vessel, an undetectable, distributed phaser system that could put a dent in a Sovereign-class starship, and shields that could potentially "punch a hole through a main-sequence star." (TOS novel: Captain's Blood)