I would go to a turn-of-the-century Boston Americans vs. New York Yankees game to get some lulz out of the hecklers in their top hats. And I ask Q to get me some potato chips from the future and bring them to our seats.
also fyi to everybody else I am talking about THE Q, not the stupid one that got stuck in the asteroid on voyager and wanted to kill himself.. that guy was a tool the REAL Q is who you'd be hanging out with.
I would ask him to, stop time and show me what is outside of our known Universe, all of it, I would want to see every single thing that exists and doesn't exist. Then when where done, we can go back to Earth, Q can click a Sovereign Class Starship (I choose this cause it looks comfy, has holodecks, plenty of room and advanced techs) into existance for me, transfer all Command Systems over to me and then maybe we hang out for a bit, I introduce Q to Jay and Silent Bob, then listen to Qs thoughts on their meaningless existence.
Oh yerr... then using my new Starship, I order all countries around the world to get along, work together and sort their Governments out or else. LoLz :p