When Q asks, "Shall we play a game?" I think it could be one of two things:

1) He is asking us to play a game with him/them, and the game is hide and seek. Q has hidden encrypted files somewhere and it's our job to find them.

2) The quote is referring to a scene in the 1983 movie "Wargames" that goes:

Joshua: Shall we play a game?

David Lightman: Oh!

Jennifer: [giggles] I think it missed him.

David Lightman: Yeah. Weird isn't it?

Jennifer: Yeah.

David Lightman: [typing] Love to. How about Global Thermonuclear War?

Joshua: Wouldn't you prefer a nice game of chess?

[Jennifer laughs]

David Lightman: [typing] Later. Let's play Global Thermonuclear War.

Joshua: Fine.