No, it's mentioned exactly once that there are twelve ships "like" Kirk's ship. The implication being that as of that episode there existed twelve Constitution class vessels. But even as early as TOS, there were more ships in Starfleet than just that class, the U.S.S. Carolina, the U.S.S. Valiant, the U.S.S. Horizon and the U.S.S. Archon being mentioned in TOS and were not Constitution class starships.
200 years before TOS "Where No Man Has Gone Before":
Captain's log, Star date 1312.4. The impossible has happened. From directly ahead, we're picking up a recorded distress signal, the call letters of a vessel which has been missing for over two centuries. Did another Earth ship once probe out of the galaxy as we intend to do? What happened to it out there? Is this some warning they've left behind?
And:
KIRK: This is the Captain speaking. The object we encountered is a ship's disaster recorder, apparently ejected from the S.S. Valiant two hundred years ago.
Since the SS Valiant was exploring 200 years earlier, it was probably not a Federation Starfleet starship like the USS Enterprise.
100 years before TOS "The Return of the Archons":
Captain's Log. Stardate 3156.2. While orbiting planet Beta Three trying to find some trace of the starship Archon that disappeared here a hundred years ago, a search party consisting of two Enterprise officers were sent to the planet below. Mister Sulu has returned, but in a highly agitated mental state. His condition requires I beam down with an additional search detail.
So the Archon was a starship type of space ship. Whether it was already a Federation Starfleet starship is unknown. Itwas more likely to be Starfleet ship since it was only half as far in the past as the first Valiant. The construction class is not mentioned.
100 years before TOS "A Piece of the Action":
KIRK: Boss? All right, Lieutenant, put him on. Oxmyx. This is Captain James T. Kirk of the starship Enterprise, representing the Federation of Planets.
OXMYX [OC]: Hello, Captain. You're from the same outfit as the Horizon?
KIRK: Yes. Unfortunately, the Horizon was lost with all hands shortly after leaving your planet. We only received her radio report last month.
OXMYX [OC]: Last month? What are you talking about? The Horizon left here a hundred years ago.
KIRK: Difficult to explain. We received a report a hundred years late because it was sent by conventional radio. Your system is on the outer reaches of the galaxy. They didn't have subspace communication in those days.
The dialog is unclear whether Oxmyx recognizes the name "United federation of Planets" or supposes that since Kirk is from outer space he must be from the same outfit as the Horizon.
SPOCK: We may quarrel with Mister Oxmyx' methods, but his goal is essentially the correct one. This society must become united or it will degenerate into total anarchy.
KIRK: If this society broke down as the result of the Horizon's influence, then the Federation's responsible, and we've got to do something to straighten this mess out. Spock, if you could get to the sociological computer, do you think you could find a solution?
Was the horizon a vessel in the Federation Starfleet, or does Kirk say that the Federation's responsible because the Horizon was an Earth ship and Earth later joined the Federation? Since it voyaged half as long in the past as the first Valiant the Horizon is more likely to have been an Federation Starfleet vessel. The Horizon's vessel type and construction class is not mentioned.
The vessel type and construction class of the Horizon is not mentioned.
50 years before TOS "A Taste of Armageddon":
SPOCK: We know very little about them. Their civilisation is advanced. They've had space flight for several centuries, but they've never ventured beyond their own solar system. When first contacted more than fifty years ago, Eminiar Seven was at war with its nearest neighbour.
KIRK: Anything else?
SPOCK: The Earth expedition making the report failed to return from its mission. The USS Valiant. Listed as missing in space.
This second Valiant was active only about half as long in the past as the Archon and the Horizon and about 25 percent as long ago as the first Valiant. This makes it more likely to have been a Federation Starfleet ship. The USS prefix also supports it. The vessel type and construction class are not mentioned.
During TOS "Friday's Child":
SULU: Warp five, sir. On course for Capella Four.
SCOTT: Warp 6 as soon as she'll take it, Mister Sulu. The captain could be in trouble back there.
UHURA: Mister Scott, another distress call from the USS Carolina.
SCOTT: Ignore it.
UHURA: The Carolina is registered in this sector.
SCOTT: Ignore it, Lieutenant. Log it as my order, my responsibility.
It looks like the USS Carolina is a real ship in starfleet. But what type of ship and what construction class isn't specified.
I believe that is all the data about those ships that might indicate whether they were in the Federation Starfleet, what vessel types they were, and what their construction classes might have been.