Well, currency can have value for many different reasons, and you're basically asking when the value underpinning the currency switched from the utility of the physical currency (i.e. the value of the metals) to fiat (this thing has value because law says it has value).

In the most literal sense, this first happened whenever the first currency got debased- since the debasement reduces the intrinsic (metal-based) value of the coin while the value of the currency keeps its elevated face value ("that law says that the coin is worth 1 Denarius no matter how little metal is in it") so it's actualy fiat, not the underlying metal (or another legitimate preference for that currency) that is determining the value of the coin.

Even Ancient Sources according to Ben Tamari seemed aware and concerned about debasement

The Mishnah and the Talmud: …bad coins buy the good ones however, good coins do not buy the bad ones” and “this is the rule, anything lesser than its friend buys its friends

Debasement definitely happened in the Roman Empire post-Augustus

In 64, Nero reduced the standard of the aureus to 45 to the Roman pound (7.20 grs.) and of the denarius to 96 to the Roman pound (3.30 grs.). He also lowered the denarius to 94.5% fine. Successive emperors lowered the fineness of the denarius; in 180 Commodus reduced its weight by one-eighth or 108 to the pound.

Severan emperors (193-235) steadily debased the denarius from a standard of 78.5% to 50% fine; in 212 Caracalla reduced the weight of the aureus from 45 to 50 to the Roman pound. They also coined aes from a bronze alloy with a heavy lead admixture and discontinued fractional denominations below the as.

But in a slightly more abstract sense, this transition from intrinsic currency to fiat didn't completely change until 1971, with the collapse of the Bretton Woods System, which had many of the world's currencies ultimately pegged to Gold - (more specifically, most currencies were pegged to the USD, which was tied to the value of gold as the US promised to sell a troy ounce of gold for $35 USD).

The BW system was untenable for a few reasons, but the proximate cause of the collapse was that the US simply did not have enough gold to cover its currency liabilities (it needed more gold to cover people who wanted to convert USD to gold), which themselves were rising because the US was printing money to cover its other rising financial liabilities (military and social spending, plus their promise to buy other currencies for a fixed price), forcing the US to abandon its commitment to purchase Gold for $35 per troy oz and thus debasing the USD with respect to the value of Gold.

Edit I may have glossed over the most interesting part. I skipped from coin-based "the physical currency is inherently valuable because it's a valuable metal" to fiat "the currency is valuable because the law gives it intangible value", but skipped over the promissory "the currency is valuable because it represents a promise in a tangible thing".

That middle section is where you see the first paper money - it's the first form of currency where the information communicated on the paper became the explicit basis for the value, as compared to coinage where the tangible value of the metal underpinned the coin's value. The first paper money was developed in the Song Dynasty and the note essentially constituted a promise by the note issuer to deliver a certain number of coins on demand. People would then trust the credit-worthiness (and legitimacy) of that promise and would trade the notes as if they were just as valuable as the coins (although some would only accept that on a discount, as they perhaps thought coins safer).

Called "flying money," it was originally authorized by the government as a draft that permitted merchants who sold commodities in Sichuan to receive payments elsewhere, thus to save them the burden of carrying the native currency of iron cash.

The interesting thing about this is that it gives us an insight into what money actually is - a promise that a reliable third party will honor on demand. Metal coins are inherently valuable so all recipients will recognize their value. Pre-fiat paper money is a (variably) credible promise to deliver inherently valuable metal on demand so all recipients (variably) recognize its value. Fiat paper is a (variably) credible promise by the state to accept that currency as satisfaction of all taxes and to force acceptance of that currency as satisfaction to all debts so all recipients recognize its value. A chequing account is a (variably) credible promise by a bank to give you fiat paper money on demand. A stock broker account is a (variably) credible promise by a broker to credit money to your chequing account... and so on.

Money isn't discrete - it's actually quite fluid. Any promise (or even any asset, which is just a set of transferable promises given by the state with respect to that aseet) can be money, as long as a third party will recognize and honour the value of that promise on demand.