The answer is probably yes, but we need to be a bit careful. Even our ancient sources generally agree that there were two versions of the story. A brief and incomplete overview of the source tradition around Alexander can be found here for context, written mainly by myself but with a supplementary comment by /u/lcnielsen on medieval Persian traditions. As with most of my Alexander answers, I'll start with laying out the sources, then take a look at each specific aspect in detail.

The event in question took place early in 333 BCE, during Alexander's campaigns in Pisidia after besieging and capturing Halikarnassos in late 334. Based on Arrian's chronology, he arrived at Gordion at some point after the death of Memnon of Rhodes, to whom is generally attributed the Persians' attempt at a naval strategy after the defeat at the Granikos in ~May 334. Memnon's death, however, had not exactly derailed Persian military efforts, and shortly afterwards the Persians regained control of Tenedos, a critical island controlling the Mediterranean side of the Bosporus. At the same time, the momentum generated by the victory at the Granikos was starting to slow down as the Persian royal army under Darius gathered in preparation to mount a counterattack in the field. As such, the moment of the solving of the knot was one at which the long-term prospects of Alexander's campaign were again coming in question.

The five surviving sources we have for the knot episode can be divided into a few camps: two 2nd century AD Greek texts based heavily on the pro-Alexander account of Aristoboulos, Arrian's Anabasis of Alexander and Plutarch's Life of Alexander; two (probable) 1st century AD Latin texts relying mainly on the critical account of Kleitarchos, Quintus Curtius Rufus' History of Alexander the Great and Gaius Pompeius Trogus' Philippic History as epitomised by Justin; and a fragment from the otherwise-lost Makedonika of Marsyas, preserved in a latter commentary on Euripides' tragedy, Hippolytos. Noticeably absent is Diodoros of Sicily's Library of History, which skips over the entire Pisidian campaign save for one particularly lurid episode which is peripheral to the point in question.

The basic approach to Alexander is invariably one of Quellenforschung (a fancy German compound word for 'source research') – why do our sources say what they do? What was the agenda of each author, and what sources were they using? For Arrian and Plutarch, they both say basically the same thing, and cite the same sources. Both first give the version that Alexander cleaved the knot apart with a sword, citing either 'some [who] say' (Arrian) or 'many writers' (Plutarch), but then also mention the account of Aristoboulos, who claimed that Alexander simply unravelled the knot by removing a wooden pin that held it in place. Neither of the Latin versions include the latter solution. Curtius, in his characteristically anti-Alexander fashion, has Alexander cut the knot in order to save face after realising he couldn't actually untie it; Justin offers little further detail. The fragment of Marsyas describes the knot itself, but not how Alexander dealt with it.

Using the major literary sources (i.e. Arrian, Plutarch, Curtius and Justin), can we establish that the Gordian Knot event took place, or at least had a basis in the primary literature? Certainly. By being able to point back to Aristoboulos for the peg claim, it is clear enough that at least one eyewitness or at least proximate account wrote about the solving of the Gordian Knot. Paradoxically, it is the story that Alexander cut the knot with a sword, despite appearing in all four sources, that is a bit more doubtful because no contemporary source can be cited for it. It is perfectly reasonable to believe it could be based on contemporary sources, and even a majority of them at that, but the surviving material just does not give any explicit indication as such. However, that Aristoboulos is our only firmly identifiable (see the note at the end) primary account does not make his version necessarily more reliable, as Aristoboulos seems to have been rather blatantly pro-Alexander, and that both Arrian and Plutarch put his version second suggests they considered his account to be of some value but possibly doubtful on this count. To quote Roller (1984), the question of how it was done 'is probably insoluble'. Still, there is no real doubt that Alexander did, one way or another, solve the knot itself.

What, however, was the significance of the solving of the knot? Hamilton's commentary on Plutarch's Life of Alexander notes an interesting discrepancy between Plutarch's account and the others – Plutarch claims the prophecy was that whoever solved the knot would come to rule the οικουμενη oikoumene, which has sometimes been taken to mean 'inhabited world', whereas Arrian, Curtius, Justin and the Marsyas fragment concur that it was just Asia (Ασια/Asia). Some 20th century scholars, particularly Fritz Schachermeyr, have argued for the possibility that in fact, the original prophecy even related exclusively to Phrygia, or at least was a largely local legend.

This relates to a more important contextual question: why was Alexander in Gordion in the first place? Arrian, Plutarch and Curtius all have it that he happened to be there because it was on a convenient route of march, and solved the knot while held up by bad weather. Only Justin has it that Alexander went to Gordion specifically to solve the knot, and given that Justin himself is most likely the latest of the writers of the literary evidence on Alexander (although the source he is summarising is somewhere in the middle of the pack), it seems to chiefly reflect later myth. This, in Roller's view, lends credence to seeing the Gordian Knot as a local, Phrygian legend that Alexander happened to be able to capitalise upon when he was there, and which ended up being inflated as time went on.

In other words, we can be reasonably sure that one way or another, Alexander solved the Gordian Knot while he was at Gordion in 333 BCE. We cannot be sure how. We can also be sure that he did not do so because he was previously aware of any prophecy relating to the knot before arriving there, and that its mythic status was generated over time, even by primary authors who accompanied Alexander, as the campaign continued and the apparent prophecy of Asian conquest became a reality.

Sources, Notes and References

1. Lynn E. Roller, 'Midas and the Gordian Knot' (1984)
2. Ernest Fredericksmeyer, 'Alexander, Midas, and the Oracle at Gordium' (1961)

Note on Marsyas: Marsyas is a weird case because his work survives only in fragmentary form, and worse still, there are two Marsyases, Marsyas of Pella and Marsyas of Philippi. While Marsyas of Pella was part of Alexander's entourage until 331 (when he seems to have returned to Asia Minor to join his uncle Antigonos Monophthalmos), we know virtually nothing of Marsyas of Philippi besides an interest in Macedonian history. While it seems probable that any references to a text called On Alexander by Marsyas is a reference to Marsyas of Pella's work, and that any to a text called the Makedonika is a reference to Marsyas of Philippi's, this is far from absolutely clear. If the Makedonika being referred to by the Scholiast on Euripides' Hippolytos was one of Marsyas of Pella, then we could be looking at a separate, identifiable primary reference to the Gordian Knot. If it was that of Marsyas of Philippi, however, then that only leaves us with Aristoboulos, as we cannot be certain of any detail as to when, why or what Marsyas of Philippi was writing.