We do not know and cannot be sure. But it seems neither likely nor unlikely, but quite possible.
It seems as if many would like to get an answer that either gives the most intimate biographical details about the bedroom behaviour of two concrete ancient persons that are dead for over 2300 years while information on both were quickly enshrouded in myths. Or an answer that simplifies things to "yeah, those Greeks did that, typical".
But for the latter case we should observe that Athens had different customs than Sparta or Thebes or Macedon, and those could change according to fashion. And the analysis of former case is hampered precisely because the story of Alexander was much more prone to be distorted because of his fame.
We cannot know for sure when looking at biographies of both party involved. But we can look at the circumstantial data of Greek/Macedon society. The sexual behaviour reported for Alexander's father Philipp may be more directly trustworthy than anything relayed to for Alexander. And Philipp was infamous in antiquity for having had a ferocious appetite in all directions.
The first thing to observe is that our concept of homosexuality is very different from how the Greeks classified sexual behaviour between for example erastes–eromenos. This ranged from something described as ritualised pederasty with clear age-limits (or, well, beard growing limits?) to genuine partnership as seen in the Sacred Band of Thebes with no such limits in age or directions of activity .
The next important clue is the specifity with which Diogenes alludes to "thighs", as in such relationships the vase paintings we have primarily show us intercrural copulation. What he might have meant by that, exactly, is object of speculation, of course. But there are a few possibilities hinted at.
We must remember the two of them had been friends at least nineteen years, if we accept Mieza as a terminus ante quem for their meeting. During much of this, they would have lived in close quarters on campaign and no doubt seen one another daily when not away on independent missions. Nineteen years is longer than many modern marriages. In terms of affectional attachment, Hephaistion — not any of Alexander's three wives — was the king's life partner. Whatever the truth of any sexual involvement, their emotional attachment has never been seriously questioned. No doubt as teenagers, both had learned from Aristotle some version of what he would later write in his Nikomachean Ethics-that perfect love was the highest friendship (1156b), and that friendship was a state of being, not a feeling (1157b). Moreover, Aristotle speaks of the friend as the 'second self (117ob) and indicates that there is only one special friend (1171a).
Thus, given the evidence for same-age homoerotic affairs in Macedonia and the weight of circumstantial testimony – even if it violates Dover's model – I do think it quite possible that Alexander and Hephaistion were physically intimate at some point. I do not necessarily think, however, that they were still physically intimate in their later years, though they may have been. Mostly, I don't think it greatly significant to the affection they held for one another.
While they may indeed have been lovers, I think it reductive to characterize their relationship solely in this way. Nussbaum (1986: 354) contrasts Greek philia with modern concepts of friendship and says that philia 'includes the vel)' strongest affective relationships that human beings form… English 'love' seems more appropriately wide ranging.'60 Greek philia could include a sexual component but extended far beyond that.Similarly, and though speaking of Achilles and Patrokios, Van Nortwick (1995: 17 18) offers an observation it would do us well to keep in mind:
We need to be careful not to misunderstand this intimacy… Friendship in general is a difficult relationship to fix. Seen in our modern cultures as existing on the boundaries of other bonds, familial or sexual, which provide the categories through which friendship itself is defined. The poems we will read here offer another model for friendship, one accommodating a greater degree of intimacy than is often accorded to nonsexual friendship these days. The first and second selves are intimate because they compose, together, a single entity… – at this level of intensity, sexual love is sometimes inadequate as a model because it may not be intimate enough. [Italics mine]
Van Nortwick's observation is a shrewd one. Our model of friendship is not consonant with theirs. Within these ancient societies where homoerotic desire was freely, sometimes emphatically, expressed, intense friendship might well develop a sexual expression even while that expression was not the focus of the friendship, or even thought of as particularly characteristic of it. 'The ancient Greeks, perhaps because their societies were so highly militarized… simply assumed the centrality of philia' (Shay 1994: 41). It would be inappropriate to refer to the friend as lover (except in very specific circumstances), as such would fall far short of encompassing the whole relationship. Alexander's choice of 'philalexandros' for Hephaistion said more about the nature of his affection than calling him merely erastes or eromenos.
Was the relationship of Alexander and Hephaistion an atypical affair? I do not believe that it was. We have shown that Macedonian society allowed same-age partnerships and seems to have accepted them without comment. Among the Pages, it was not only possible, but perhaps even to be expected that young men would form friendships with one another that included a sexual aspect, but was not limited to it.
What, then, might we gather from this detailed look at one example? Simply that models – even good ones based on careful analysis of the evidence – can put blinders on subsequent scholarship – if we are not careful. Without the cognitive dissonance created by the sheer bulk of circumstantial testimony in the case of Alexander and Hephaistion, it would be easy to overlook a relationship like theirs. Even with the circumstantial evidence, we still cannot be at all certain they were lovers. Because such relationships are not atypical for their societies, sly insinuations – such as those made about Agathon the Tragedian – are absent.
It does cause one to wonder how many other such relationships may have existed between less famous philoi. While the Dover Model describes the most common-and least ambiguous-form for homoerotic expression in ancient Greece, it was not the only one. There were other options, particularly in military contexts. In short, a confusion of terms may make it difficult for us to pinpoint other such typical 'atypical' affairs, and we should take this into account when employing our models. All such models are to some degree artificial constructs; we should expect them, then, to be ultimately inadequate.
— Jeanne Reames: "An atypical affair? Alexander the Great, Hephaistion Amyntoros and the nature of their relationship", History Faculty Publications, 17,, The Ancient History BulletinVolume 13, Issue 3, 81–96, 1999. (PDF)