Yes, that's possible:
$("#id-of-button").click(function() {
var bg_url = $('#div1').css('background-image');
// ^ Either "none" or url("...urlhere..")
bg_url = /^url\((['"]?)(.*)\1\)$/.exec(bg_url);
bg_url = bg_url ? bg_url[2] : ""; // If matched, retrieve url, otherwise ""
alert(bg_url);
});