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In Star Trek IV: The Voyage Home, we have this exchange:
McCoy: It's a wonder these people ever made it out of the 20th Century.
Kirk: They're still using money. We've got to get some.
And this exchange:
Dr. Gillian Taylor: Don't tell me they don't use money in the 23rd Century.
James T. Kirk: Well, we don't.
Are those the first explicit mentions of the lack of currency in the Federation/future? Was there anything regarding that in TOS, TAS, or movies I-III?
From "Arena" Production # 19, aired 19 Jan. 1967:
KIRK: A large deposit of diamonds on the surface. Perhaps the hardest substance known in the universe. Beautifully crystallized and pointed, but too small to be useful as a weapon. An incredible fortune in stones yet I would trade them all for a hand phaser, or a good solid club. Yet the Metrons said there would be weapons, if I could find them. Where? What kind?
"Mudd's Women" production # 4, aired 13 October 1966:
COMPUTER: Offense record. Smuggling. Sentence suspended. Transport of stolen goods. Purchase of space vessel with counterfeit currency. Sentences, psychiatric treatment, effectiveness disputed.
And:
MUDD: Oh, you beautiful galaxy! Oh, that heavenly universe! Well, girls, lithium miners. Don't you understand? Lonely, isolated, overworked, rich lithium miners! Girls, do you still want husbands, hmm? Evie, you won't be satisfied with a mere ship's captain. I'll get you a man who can buy you a whole planet. Maggie, you're going to be a countess. Ruth, I'll make you a duchess. And I, I'll be running this starship. Captain James Kirk, the next orders you're taking will be given by Harcourt Fenton Mudd!
In "The Menagerie, Part 1", production # 16, aired 17 November 1966:
PIKE: To the point of considering resigning. BOYCE: And do what? PIKE: Well, for one thing, go home. Nice little town with fifty miles of park land around it. Remember I told you I had two horses, and we used to take some food and ride out all day. BOYCE: Ah, that sounds exciting. Ride out with a picnic lunch every day. PIKE: I said that's one place I might go. I might go into business on Regulus or on the Orion colony. BOYCE: You, an Orion trader, dealing in green animal women, slaves?
Of course that scene was not actually filmed for "The Menagerie, Part 1". It was a scene from "The Cage" the first pilot film, reused in "The Menagerie, Part 1". So although it was seen somewhat later than the one in "Mudd's omen" it was filmed in the earliest Star Trekproduction of all.
The TOS Episode "The Trouble with Tribbles" explicitly mentions spending "credits" on tribbles. Cyrano Jones is clearly an entrepreneur of sorts, so it's likely that "credits" are actually some form of currency. Perhaps the TOS Federation economy is similar to that of East Germany or the USSR, where basic necessities are provided for free or for nominal costs while luxury goods (which could include tribbles or other exotic pets) are rather expensive.
It's also possible (but less likely) that "credits" are actually some form of currency-like accounting system, perhaps a measurement of the amount of stuff that a crew member is allowed to store in their Starfleet crew quarters.
Later series mention "transporter credits" in the currencyless 24th century Federation (e.g. cadets at Starfleet Academy using them to beam home to visit parents), so it seems that by then the main limiter in terms of wealth is the cost of moving people and stuff around, not the cost of making doodads.
Apart from Robert Columbia's TOS reference there were a few mentions of "credits" in the second and third movies according to this Wikipedia article:
In other episodes, especially earlier in the in-universe timespan, a monetary unit known as the "credit" is mentioned.
At the Federation space station K-7 in the original series episode "The Trouble with Tribbles", set in 2267, Uhura is offered a Tribble for 10 credits.
In Star Trek III: The Search for Spock, in 2285, while on Earth, McCoy attempts to hire a ship to take him to the Genesis Planet, is warned it would be expensive, and haggles over payment; we do not know if McCoy could have afforded this or how much it would cost, since he was taken into custody for breaching the secrecy of the Genesis Project immediately afterwards.
And in Star Trek II: The Wrath of Khan, Carol Marcus mentions the Federation's decision whether or not to "fund" the Genesis Project itself, though "fund" means something different in this context as credits are not mentioned.
In the Deep Space Nine episode "You Are Cordially Invited...", Jake Sisko tells Quark he sold his first book, but when Quark asks him how much he earned, Jake answers, "It's just a figure of speech." This explains moments when characters have made similar comments (in Star Trek VI: The Undiscovered Country, for instance, Scotty mentioned having "bought" a boat, and during the film Star Trek Generations, Captain Kirk states that he "sold" his house).
The first mention of money in TOS was certainly Sulu in Where No Man Has Gone Before.
Here everyone is in the (unusually large) conference room to discuss the growing menace of their crew mate as his ESP powers are increasing. Sulu says,
Speaking mathematically his powers are increasing geometrically. As if you take a penny and double it everyday, in a month you'll be a millionaire.
It always disappointed me that they took us all the way into the future and still had to go back for a twentieth century reference like that.
This being I think the second pilot may fall behind the reference in The Cage but it is the only mention I think of actual coinage in all of Star Trek.
I think the dialogue mentioned in the question is in regards to actual physical currency like paper dollars and metal cents. Physical money would be inconvenient in an advanced society. There would be more convenient, secure and transparent accounting systems. Even our early twenty-first century technology makes cash almost obsolete.
Except for New York, most people under the age of 60 do not carry cash in the United States. Even fewer people carry checks. In the recent Covid-19 pandemic, almost all businesses in the U.S. are refusing transactions where cash has to change hands. Credit cards are ubiquitous. Beyond that, there is Apple-pay, Google-pay, Android-pay, PayPal, Zelle, Venmo, etc. Extrapolate that out to a society that has interstellar travel amongst different planets and species, hard currency would be impractical. We have enough problems now with one planet having various dollars, various pesos, various pounds, euros, yen, rand, rial, rupees, and the other over one hundred denominations of currency.
