I'm not sure about citation in other traditions, but in the Greek world it goes back to at least the fifth century BCE.
The Histories of Herodotus (writing around the 420s BCE) might qualify as the earliest to cite an ancient author for an argument. He quotes from the Iliad (6.289-92) and the Odyssey (4.277-30 and 4.351-352) in order to argue that Homer knew of the tradition that Helen in Troy was actually a phantasm, and that her real body was in Egypt.
Herodotus actually quotes from others, too, including famously Pindar's line that "nomos is king" (Hdt. 3.38.4), though the usage is less of a scholarly argument and more of a relation of an aphorism. However, Plato in the Gorgias (written around 380) in fact cites the very same passage in order to strengthen an argument. He places the quote in the mouth of Callicles, a sophist, who says:
δοκεῖ δέ μοι καὶ Πίνδαρος ἅπερ ἐγὼ λέγω ἐνδείκνυσθαι ἐν τῷ ᾁσματι ἐν ᾧ λέγει ὅτι
—"νόμος ὁ πάντων βασιλεὺς / θνατῶν τε καὶ ἀθανάτων."
And this I take to be the sentiment of Pindar, when he says
in his poem, that:
Law is the king of all, of mortals as well as of immortals;
Whereas Herodotus cites Homer on a very limited basis (and some have argued that 2/3 of those citations are spurious), it's more systematic in Plato, showing an in-depth engagement with literary sources. More than Herodotus, I'd say Plato is the first to truly cite others.
The approach is more formalized in the Hellenistic era. With the construction of the libraries at Alexandria, Pergamum, and Antioch, scholarship on texts depended much more heavily on accurately citing those texts. It is in this milieu where we get the scholia and those traditions. They're citing not only words, but sometimes book numbers and naming poems. Their citations are closest to the modern ones. As Eleanor Dickey puts it:
The real beginning of Greek scholarship in our sense of the term, however, occurred with the foundation of the library and Museum at Alexandria in the early third century BC, and for centuries the librarians and other scholars there were the most important Greek scholars.
At this point, though, we're nearly two centuries after Herodotus' Histories.
So the answer to your title question would be "the Alexandrian scholars" and the answer to the question in your body would be "probably Herodotus, at least by Plato."
For general works on Greek scholarship, I'd say start with two books: Eleanor Dickey's Ancient Greek Scholarship and Rudolf Pfeiffer's The History of Classical Scholarship. The latter is more of a general history of scholarship, while the former is more geared towards a modern scholar who desires access to scholarship of the ancients.
For specific scholarship on Homeric criticism, you might want to start with the Wikipedia page on Homeric scholarship, which has some numbers on Homeric quotations. There's a newer monograph on Herodotus specifically, Bruno Currie's 2021 Herodotus as Homeric Critic, but I've not yet had a chance to explore it in depth.
If you have access to JSTOR/ILL, two articles that discuss the quotes I mention at length are Neville's 1977 "Herodotus on the Trojan War" (G&R 24.1: 3-12) and Humphrey's 1987 "Law, Custom and Culture in Herodotus" (Arethusa 20.1: 211-220). I also found this talk, whose abstract might interest you. And for Plato and Pindar, and the lyric poets in general, see Demos' 1999 Lyric Quotation in Plato. There's a lot of scholarship on Plato's citations of other works, but this one would get you started, and it's free to read online.