[ SPOILER ALERT ] - the following is from endings of Gundam Wing, Endless Waltz, and various Gundam Wing mangas.
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No one knows what the future will hold. Luckily, after Endless Waltz ends, it really isn't the end yet. There is a manga that follows after the Endless Waltz movie called Ground Zero. (I've never read it so can't say anything much yet) Also before EW, another manga called Blind Target led up to it. Through the mangas and Endless Waltz, we have a rough idea of what Quatre does after the anime series. As we all know, we saw Quatre got stabbed by Dorothy and was rescued at the end by Trowa. Though the series did not show a full ending about all the pilots, that was why the movie was made.
odd & even numbers
I recently found out about a mini movie called "Odd and Even Numbers" that talks about what exactly happened after Libra was shot to dust. The gundams returned to Peacemillion and Quatre was sent to a hospital right away I guess. The clip of him starts in a hospital with Rashid beside him. He talks about how the war is finally over. At the very end, you see Trowa, Duo, and Quatre celebrate with champagne... glasses as they promised.
The Endless Waltz movie talked about what happened after the war in A.C. 195. Everyone believed they were down the road to peace once again, but it did not go that way at all. Mariemaia, Treize's daughter wanted to rule the world. It failed to succeed because Heero and co. got their gundams in time to stop her. After the retaliation ended, it showed that all the pilots' gundams were self-detonated. Quatre was seen at the ending song among the Maganac Corp. with spacesuits on. So I suppose they're on a spaceship. I believe they're heading back home, where the L4 colony is. Also, there is a little part after the sequel. It showed Quatre with a contruction worker's hat. Rasid was handing a mug to him. (maybe tea again?) It seems like Quatre was carrying the plans to a building or something. They were just rolls of paper so I guessed. ^_^ Perphaps they were rebuilding some parts of a colony? Not necessary L4 but it's a big possibility that it might.
Blind Target's timeline came before Endless Waltz. In this manga, it already showed Quatre as a peace ambassador and head of the Winner family. Going to different colonies and places to have peace talks. Being the heir of the Winner family, he is still highly respected. He's even wearing a suit. (of course he is, why wouldn't he? =P) My guess is that he has a position similar to Relena's.
In the Episode Zero manga's Preventer 5 short chapter, it showed Quatre as a very busy man. He had piles of paper on his desk which are documents for him to read. He seemed like he was tired of doing this kind of work but can't give up because he didn't want others to do it for him, or maybe he just doesn't want people doing his work for him.
During the series, he became well known when he inherited his father's property, including money and being the heir of the Winner family. That is a big responsibility for a 15 year old kid! That's not all though. After the Endless Waltz movie, he had become a Preventer 5 supposedly. This is reinforced by the fact that there was a Preventer 5 chapter after Episode Zero. It showed Quatre planning all the battle strategy. He looks pretty tired, but when Wufei asked for his help, he jumped up and quickly went with him to see the other pilots. Even though Quatre has so much work to do, he never gives up as long as he can help someone when they need help. In Blind Target, it also showed him going to conference meetings. I believe he is a peace ambassador as well. Similarly, he does what Relena does. To talk of peace in conference meetings to discuss new plans of immigration and building projects. Maybe that is why Quatre is shown at a construction site. He might be already in charge of a project. I think he wants to help a lot of people but just can't because he is already feeling tired. For a 15 year old young leader with massive amounts of assets, he is very GOOD! Amazing! I don't know how to describe it but it just makes one feel so proud of it. Like he deserved all the recognition.
As for dreams, we will never know. =D We do know that he wants nothing more than peace. The world has already acheived that so I think he might have a new goal. I don't know for sure, but I know there might be a chance he might start a family, though he already has a family with the Maganac Corps. He does look lonely since everyone is viewed with a girl in the endings. (no offence to Trowa) But I see that Quatre is not the kind of person to abandon his work for love.