If it wasn't an exchange, I assume you didn't send anything in exchange for bitcoins, right? You just received them? How? You must remember something. Did you perhaps win a competition in that online game? There was a game with roots in 2010, someone had tweeted that once, but I can't find it. There were also some rewards in another multiplayer game. I think first two earned 100$ and the third 250 BTC? Sort of.

If you didn't install Bitcoin-QT at that time (2009-2010) it's highly unlikely you ever really owned bitcoins.