You could add the image to the quiz array like so:
var quiz = [{
"question": "The first question doesn't seem to be showing?",
"choices": ["Don't", "Know", "Why"],
"correct": "Why",
"image": "http://upload.wikimedia.org/wikipedia/en/thumb/f/f4/The_Scream.jpg/475px-The_Scream.jpg"
}, ...]
Then add a line in the askQuestion function that swaps out the src attribute of the image based on the image set in the array above:
$("full-screen-background-image").src = quiz[currentQuestion].image;
I've knocked up a quick JSFiddle based on the code you provided http://jsfiddle.net/69BPA/