Did ancient Greeks harden steel weapons?

Yes. They did harden their weapons, not always, that is from the start of us calling them "Greeks", but from quite early on, and certainly in the timeframe of the question.

But the Hittites are credited with learning to make a metal hard and tough enough so that it could be used for knives and swords. They did not know that they obtained this result through subjecting the metal to the influence of carbon monoxide long enough so that they had not only reduced the iron oxide to metal but had caused the metal to take up enough carbon so it became the iron-carbon alloy which we call steel. All they knew was that if they did it the right way they got a metal that was superior in many ways to anything previously produced. We do not know, nor does it really matter, whether they secured the reduction and carbon absorption in a single operation, or whether they first made the metal and then caused it to absorb carbon in a subsequent heating. Probably they did both, learning from the first that they could make a metal so hard and tough as to be very useful, but also one that was very hard to work. By working the metal into the desired shape while it was still soft and then causing it to take up the hardening carbon, the process was facilitated.
_{–– Thomas T. Read: "Metallurgical Fallacies in Archaeological Literature", American Journal of Archaeology, Vol. 38, No. 3 (Jul. - Sep., 1934), pp. 382-389.}

For a more general overview, with a focus on material and technology:

E.A.Ginzel: "Steel in Ancient Greece and Rome"

Keywords: Refutes claim the old ones did not know steel, Fe smelting known well around 1000 BC, ref. to Homer, Aristotle, Herodotus, maximum temperature, Romans around 1100oC, smelting = solid state conversion, table bloom composition, definition steel varieties,

For a look into the ancient sources, slightly outdated:

With respect to steel, it is a much disputed point whether this metal is the proper sense of the word κύανος in Homer (Il. XI.24, 35, Od. VII.87) and Hesiod (Scut. 143), but at all events it is highly probable that this is the meaning of ἀδάμας in Hesiod (Scut. 231, Theog. 161; see the lexicographers, s.vv., the commentators on Homer and Hesiod, in ll., and Beckmann, vol. II p324). It would appear from the manner in which Aeschylus refers to the Chalybes, taken in connection with the traditions respecting the early intercourse of the Greeks with the shores of the Baltic, that the iron and steel works of that people were known at a very early period, and that it was from them chiefly that the Greeks procured their iron and steel (Aesch. Prom. 720; Apollon. Rhod. II.1000; Xen. Anab. V.5 § 1; Ritter, Erdkunde, vol. II p776; Höckh, Creta, vol. I p294).

Enough has already been said respecting the early knowledge of the precious metals, separately and in combination, under Argentum, Aurum, and Electrum. In drawing inferences, however, from Homer's allusions to these and the other materials of the useful and fine arts, we must be on our guard not to make the poet's imagination our standard of their actual abundance. (See further, concerning the real or supposed knowledge of metals and metallurgy in the earliest times, Plin. H. N. VII.56 s57.)

If we turn from the metals themselves to the art of working them, still taking the poems of Homer and Hesiod for our guide, we find the Greeks of that early period perfectly acquainted with the process of smelting the metal from the ore and of forging heated masses into the required shapes, by the aid of the hammer and tongs. It may, indeed, be doubted whether the χόανοι, into which Hephaestus throws the materials of the shield of Achilles, and which are worked by the blast of twenty pairs of bellows (φῦσαι) are smelting-furnaces or mere smith's forges (Il. XVIII.470), but the former sense seems to be required in the passage of Hesiod (Theog. 863).

Both Homer and Hesiod refer to the smith's workshop (χαλκήϊος δόμος, χάλκειος θῶκος) as a common lounge and as a place of shelter to which the poor resorted for its warmth (Od. XVIII.328, Op. et Di. 491). The whole of Homer's description of the workshop of Hephaestus deserves careful study (Il. XVIII.369, &c.). The smith's instruments were the anvil (ἄκμων) with the block on which it rested (ἀκμόθετον), the tongs (πυράγρη), and the hammer (ῥαιστήρ, σφῦρα, Il. l.c., Od. III.433‑435). [Incus, Forceps, Malleus.] The arts of casting metals into moulds, and of welding, or even of soldering pieces of metal together, were as yet unknown. In large works, hammered plates were united by mechanical fastenings, nails, pins, rivets, cramps, or dovetails (δεσμοί, ἧλοι, περόναι, κέντρα), and specimens of this sort of work in the bronze statues p760 of the earliest period were still to be seen in the time of Pausanias (Il. XI.634, XVIII.379; Paus. X.16 § 1).

The art of embossing, or fastening pieces of one metal on to the surface of another (ἐμπαιστικὴ τέχνη), is referred to several times in Homer (Il. XI.24, 35; Lobeck, ad Soph. Aj. 846, &c.). Gilding was commonly practised; one interesting example is the gilding of the horns of an ox about to be sacrificed (Od. III.425, &c.). This passage furnishes a striking instance of the use of words connected with χαλκός for working in any kind of metal: thus, the artificer is called by the generic term, χαλκεύς (432), as well as by the specific name, χρυσοχόος (425), and his tools are the ὄπλα χαλκήϊα, οἷσίν τε χρυσὸν εἰργάζετο (vv. 433, 435). Lastly, the images used to describe the hissing of the burning stake when plunged in the eye of Polyphemus, shows an acquaintance with the process of dipping red-hot iron in water to harden it (Od. XI.391, comp. Soph. Ai. 720).
_{–– Philip Smith: "Metallum", in: William Smith: "A Dictionary of Greek and Roman Antiquities", John Murray: London, 1875.}

Geographic and geologic prerequisites and including a synthesis of both of the above to be found specifically for Greek metallurgy in:

Smelted iron has been known since the third millennium B.C. (Coghlan 1956: 61 sqq.). The Minoan and Mycenaean civilizations used iron for amulets and jewellery (rings for instance) (Iakovidhis I970: 288–96). Towards the end of the Mycenaean period and in the beginning of the Dark Ages the Greeks began to use iron for weapons: knives, daggers and swords (Snodgrass 197I: 217 sqq.). In the Near East iron became widespread by 1200 B.C. The iron dagger blade from the tomb of Tutankhamen (c. 1350 B.C.) and the iron blade of the battle-axe from Ras Shamra (c. 1300 B.C.) are top pieces of the iron metallurgy of this age (Singer 1954: 619). The techniques of hot-forging (hammering) and carburization had been mastered now (see e.g. Coghlan 1956: I34–5I). Iron objects with hard and sharp cutting edges or points could be made in this way.

Around the middle of the eleventh century the Greeks themselves may have begun making iron swords, knives, daggers and pins, from local ores (Snodgrass 197I: 221 sqq.). For the time being iron did not yet surpass bronze in all respects. The hardness of the entire objects was often less than that of bronze objects, and the toughness of later steel objects was unknown as yet. For about one or two centuries no great technological progress seems to have been made. It is with new techniques – quenching, and piling together thin laminations of carburized iron – that the quality of iron (steel) objects increases in such a way as to make a lasting impact on technology (it is not known how early the technique of tempering became known; Wootz steel and pattern-welded true damascene steel [Coghlan 1956: I55–65] were not yet fabricated by the Greeks).

From the ninth or eighth centuries B.C. onwards iron of great hardness and steel of fair elasticity could be fabricated (pace Keene Congdon 1971). The variety of iron utensils increased considerably then (Pleiner 1969: i8).

It is recorded for c. 600 B.C. that Chalcis was a well-known production centre of swords (Alcaeus F357Lobel and Page, 1.7; cf. Aeschylus F703Mette, Callimachus F236Pfeiffer, Hyginus, Astr. II, 6; see also Archilochus F3Diehl). The blades of these swords must have been of steel.

Central Euboea is rich in iron ore deposits, see fig. i (Anastopoulos 1960; Melidh6nis 1960; Katsikatsos 1969a; Katsikatsos 1969b; Katsikatsos 1975). The great majority of the Euboean iron deposits lies in the territory of ancient Chalcis. The Chalcidian iron and steel industry may have used local ores. Euboean iron mining is recorded in ancient authors (Callimachus F70o, Pfeiffer; Strabo X, I, 9, p. 447C; cf. Heraclides Lembos F62Di)
_{–– S. C. Bakhuizen: "Greek Steel", World Archaeology, Vol. 9, No. 2, Architecture and Archaeology (Oct., 1977), pp. 220–234. (jstor)}

What better way than to close with Homer:

It is worth mentioning the blinding of Polyphemus by wily Odysseus described by Homer in Od. 9.390-394):

. . . As a blacksmith plunges a hatchet into cold water to harden it – the latter makes a great hiss, giving strength to iron - even thus did the Cyclops’ eye hiss round the beam of olive wood . . .

^{Two small iron daggers from Perati, Attica.}
It is worth mentioning some iron weapons, that the author had the opportunity to study from a metallurgical point of view some years ago. Among them, two small iron knives of 12th century BC (Figure 7), found at Perati by S. Iakovidis, Professor of Archaeology and distinguished member of Athens Academy. No similar items, belonging to this remote era, were found.

The present research shows that iron metallurgy was greatly developed during the geometric period and influenced Homer to such an extent, that he refers very often to this metal either as an indication of wealth, or as an important material for hard tools and weapons. What is more, Homer was impressed by its magical property to harden when (a) transformed into steel by carbonization and (b) subject to heat treatment of the latter. No other metal possesses these unique properties.
_{–– George Varoufakis: "Iron in the Homeric Epics & Homer, a Sensible Ecologist", in: S.A. Paipetis (Ed): "Science and Technology in Homeric Epics", History of Mechanism and Machine Science Volume 6, pp 103-111, Springer: 2008.}