"Q-ship" in the Honorverse refers to ships which appear to be merchantmen but are actually armed merchant cruisers. There are also armed merchant cruisers which do not attempt to disguise what they are.
In either case, she is usually a ship which has been fitted out with weapons and defenses on a standard merchant hull. The armed merchant cruiser is regarded as a cheap stand-in for a real warship to be deployed where nothing really nasty is expected to come calling. Frequently, the AMC has greater endurance than a warship of equal force (not to be confused with a warship of equal tonnage) and so may be deployed for purposes such as convoy protection in place of a destroyer or a CL. Most AMCs, as most "Q-ships", are not really intended to stand up to heavy warships. If they have to, then they have crapped out, and they usually get blown out of space, as happened with Honor's ship in Honor Among Enemies.
A "Q-ship" may be thought of as a stealthed AMC. It's function is to get into range for ambush attacks, whether as a convoy escort or as a convoy raider. As such, disguise is at least as much a part of its design function as armament. Honor's "Trojans" in Honor Among Enemies were employed in precisely this fashion. Note that it is sometimes considered useful to actually warn raiders, if they are operating in light ships, that "Q-ships" have been deployed to their area. Since they can't know when their target is a real merchie or a disguised AMC, they tend to operate much more cautiously or even leave the area to seek new space in which to cruise. If heavier opposition is expected -- as in, for example, a Peep warship acting as commerce-raider -- then it is obviously most advantageous for the "Q-ship's" presence to be unexpected until it opens fire.
Some "Q-ships" are actually armed naval auxiliaries which were built as such from the keel out. These vessels normally have military grade compensator and nodes and better sidewall generators than most merchant conversions. They are not, by any stretch of the imagination, regular warships, their complements tend to be much smaller, they are far cheaper to operate and maintain, and their armament -- by the standards of a warship their size -- is extremely light. In short, they are merchantships with military propulsion and somewhat tougher defenses, but not exceptionally so. They are also more likely to be recognized as what they are, since their emission signatures, hull form, design features, etc., are often well known and give them away. To use a modern wet-navy example, it is very difficult to confuse one of the USN's purpose-built underway replenishment vessels as a civilian tanker.
The Peeps, unlike most other navies, also designed and built "Q-ships" that were, in effect, military hulls. One of these was Sirius in On Basilisk Station. These ships were close to the size of an SD, but they were never intended to serve as SDs and they carried no more armament than a BC or maybe (in some cases) a BB. In Sirius' case, the missile armament was somewhat greater than that of a BC, but the beam armament was less than that of most CAs. However, the ship had military framing and compartmentalization (hidden inside the cocoon of her cargo holds), military compensator and impellers, and all-up military-grade sidewalls, sensor suite, and EW/ECM capability. She was not built as a commerce raider. She was built as part of the Peeps' conquest strategy. Her design function, and that of her sisters, was to pose as merchant ships -- complete to hauling cargo -- with their armament concealed from all but a very close inspection, in order to infiltrate the critical defensive areas of the next single-system polity on the Peeps' list. They were, in a very real sense, Trojan Horses, whose function was to hit the defenders from behind or to initiate the attack by sneaking in close and taking out critical system defense installations or even warships in parking orbit around a planet. In short, they would be used for the raiding functions a BC might have been employed upon, but using stealth, rather than speed, to get into attack position.
Obviously, they had other functions, and the Peeps' Special Ops people were delighted to get their hands on them for operations like the one in On Basilisk Station. Of course, using one of them posed problems of its own, as was demonstrated in On Basilisk Station when the wheels came off. They could also be used for any of the "Q-ship's" other normal missions: commerce raiding, commerce protection, system picket, etc.
They were considerably more expensive to build than it would have been to convert an existing merchie hull -- or even a typical armed auxiliary -- with the same armament. They were not much more expensive to operate than the conversion would have been. They were much tougher and capable than the conversion -- for example, what happened to Honor's ship in Honor Among Enemies would not have happened (or not, at least, so catastrophically) to Sirius under the same circumstances.
The rationale for building such units is, obviously, that there is a special mission waiting for them. The Peeps used them successfully in taking down several of the systems they conquered, including Trevor's Star. In at least two cases, they used only "Q-ships" and took out relatively weak opponents without fully mobilizing and without giving their intended targets (or anyone else -- like Manticore -- who might have protested) time to mobilize (or intervene). As such, the ships well repaid their initial cost. Once their existence became generally known, they became less effective and were relegated to other roles, although, of course, they always remained available for their originally intended missions.
Under normal circumstances, Sirius should have taken Fearless. And, indeed, she effectively did, except for the fact that Honor was waiting for her with the grav-lance when she closed to finish Fearless off. As the engagement actually broke down, Sirius discovered that her EW capabilities were far inferior to those of Fearless (particularly critical in a missile duel), which offset much of her advantage in firepower and size. She remained much tougher than Fearless, and far more capable of absorbing damage and remaining in action, however. Another huge factor in the way the engagement developed was that her skipper literally could not afford to be identified or to remain in the system. His entire mission, from the moment the ground-side operation went wrong, was to get out of the system. It was not to destroy Fearless; it was to minimize the scale of the disastrous incident the operation had become and, above all, to keep anyone from knowing that the PRH had been officially involved. In regard to the last point, it was also critical that he not submit to Honor's demand that he stop to be boarded and examined, since that would have absolutely established Peep involvement in events in Basilisk. The best way for him to accomplish his mission (or, rather, to accomplish what his mission had become) was to get away into hyper without fighting Fearless at all.
As a result, he began by trying to run, and but for the alertness of Honor and her crew (and especially Honor) would probably have succeeded in doing so without interception. Once Honor went in pursuit and was in position to overhaul, he hoped to fight a running engagement in which his superior chase armament would allow him to inflict crippling damage on her ship. What he did not want to do, given his weak energy armament, was get into a beam engagement with her... since he didn't know what had been done to Fearless's energy weapons. Had he possessed equally good EW and missiles, he would have won easily. Even with the much inferior EW and missiles he actually had, he would have won in the end (with only moderate damage) but for the grav-lance.
The real irony (and tragedy) of that entire battle was that Honor was sacrificing her own ship, knowing she was doing so, against a vastly more powerful opponent in an effort to prevent Sirius from bringing in the Peep fleet... while the Sirius was desperately trying to break off and run in order to tell the fleet not to come.
But the real point here is that when Honor & Co. referred to Sirius as a military-built design, they meant just that. They were not referring to a converted merchantman or even to a fleet auxiliary. In effect, they were talking about an SD-sized BC with standard cargo holds wrapped around the core military hull in order to facilitate its disguise as a merchantman. In no sense, however, could she have been considered equivalent to a regular warship of her size, nor did she cost any appreciable fraction of what a BB or an SD would have cost, either to build or to operate.