Do we have any non-Greek mentions of the Trojan War?

Yes, plenty. There are depictions of scenes from the myth in Etruscan art starting in the 8th-7th century BCE; there's evidence to link characters from the legend to Roman traditions starting in the 5th century BCE (or even 6th, if you really push it); and of course once you get to the mediaeval period, the legend pops up in dozens of non-Greek traditions.

It may be that you're asking about contemporary evidence for a supposed historical event, though. In that case the answer is no; though there are two texts that a handful of scholars like to bring up anytime the possibility of a Trojan War is mentioned.

One is a letter written by a 13th century Hittite king (identity uncertain) to the king of Ahhiyawa, which was probably a Greek kingdom of the time (location unknown). Here's Beckman's translation from The Ahhiyawa Texts (SBL, 2011); in the first couple of lines the Hittite king is asking the Ahhiyawan king to give an order to a fugitive that has taken refuge in Ahhiyawa. "Wilusa" is the Hittite name for Troy.

(Please say this to him:) "...The King of Hatti has persuaded me about the matter of the land of Wilusa concerning which he and I were hostile to one another, and we have made peace. Now(?) hostility is not appropriate between us. ..." ...(a few lines omitted)... And concerning the matter [of Wilusa] about which we were hostile -- [because we have made peace], what then? If [a certain ally] confesses an offense before his ally, [because he confesses] the offense before his [ally], he does not reject [him.]

And here's Trevor Bryce's commentary on the passage (p. 121, same volume):

This is the only occasion in the Ahhiyawa corpus where there is a reference to what appears to have been direct conflict between Hatti and Ahhiyawa. In all other cases, hostile action by Ahhiyawa against Hatti appears to have been limited to support for the activities of local insurrectionists like Piyamaradu. However, we do not know what the nature or the scale of the hostilities was on this occasion, whether it amounted to outright war, a skirmish or two, or merely a verbal dispute conducted through diplomatic channels. (The verb ku-ru-ri-iḫ-ḫu-e-en used in this context could mean any of these things.)

So if you really want you can treat this as a possible reference to a heavily distorted version of the legendary war. But no one who didn't already have the myth at the back of their mind would see this as a reference to a war, let alone a ten-year war between a united Greek mainland and an alliance of Anatolian states. Note also that (a) the Hittite king casts the Hittites as the aggressors in the dispute, and is trying very hard to apologise; (b) Troy is a formal part of the Hittite empire at this date, but no Hittites, or any other client kingdoms of the Hittites in western Anatolia (except one, Lycia), appear in any of the mythical sources.

This is typical of the kind of way you have to make excuses for the evidence if you want a historical Trojan War. In a similar fashion, we have the text of the treaty that incorporated Troy into the Hittite empire, which names the king of Wilusa as Alaksandu -- very likely a hittitised form of "Alexandros". But again, that's not consistent with mythical sources (the mythical Alexandros was never king), and Alexandros is a common Greek name.

So really how you read the evidence depends on your motivations. If you're trying to rationalise a Trojan War as a historical event, you'll be strongly motivated to dismiss the inconsistencies and problems in these documents; and it's not actually impossible that you'd be right in doing so. Certainly there are a handful of prominent scholars who think this way. But they are very much a minority. If you don't start with that motivation, there's nothing in the Bronze Age documents that would indicate anything resembling the legendary war to someone who wasn't out to rationalise it.