The Wooden Horse -- Trojan Horse is technically a misnomer, even if we use it commonly -- was ostensibly an offering to Athena, to compensate her for the Greeks' sacking of her temple (one of many smaller episodes that occurred over the course of the long Trojan War), and to ensure a safe return to Greece.

It was not intended to be a gift to the Trojans, though it's obvious why you'd think that (i.e. the whole timeo Danaos et dona ferentes thing: "I fear the Greeks, even when they bring gifts"). However, a Greek spy, Sinon, was left behind and he managed to convince the Trojans to take the horse as a trophy into their city anyway -- despite naysayers like Laokoōn (who was killed by a serpent to ensure the success of the Greek plan) and Kassandra (who was cursed to never be believed by anyone).

"Post-war gift-giving" is a little vague. There is nothing that springs to mind that recalls the instance of the Wooden Horse (not a gift!), but perhaps some other redditors can offer concrete examples of what you are looking for? Obviously, there are examples of one side offering reparations to the other: there is an example of sorts in the Iliad when the Trojans discuss giving Helen back to the Greeks, along with a great deal of movable wealth (because Paris didn't just take Helen when he left Sparta!), in order to pay off the Greeks and have them return home.

For details on the Wooden Horse, see the Odyssey (where there is the story, early on, of Helen mimicing the voices of the wives of the Greek men inside the horse), Quintus Smyrnaeus' Posthomerica (for some of the finer details), and especially book 2 of Vergil's Aeneid (for the fullest account). I also wrote about the Wooden Horse on my website, for example this article on the Mykonos relief vase, one of the earliest depictions of the thing.