It's technically simple: Go to that player's personal page. In the top row you can find a button "Send friend request". Klick it (at any time: during the game, after the game, the next day...) - that's it. If they accept, you will get a notification. After that, you'll get a notification, when that player goes online, or a status info next to your avatar, if they are online, when you come to BGA.
If they don't accept, you won't get a notification. I guess chances are higher they'll accept, if you had some chat together during or after the game. You might just ask: "Hello! Would you like to play this game more often? May I send you a friend request?"
