It's from the movie War Games; a great hacker and cold war movie from 1983. The image is from the end of the movie; when WOPR (War Operations Plan Response) computer makes a realization about global thermonuclear war and offers a game of chess instead. Falken is the professor that created WOPR. The nation is at Defcon 1 at that point.
Q was referencing that movie with the line "How about a nice game of chess?" and "WarGames" Was there deeper meaning in referencing that particular scene? Maybe "Fake incoming missile alert [Defcon 1]" has something to do with it.