The simple thing you can do is to make HEAD request, so that you don't actually download the file but only response headers. From there you get Content-Disposition header which contains filename field.

Something like this in jQuery:

$.ajax({

    type: HEAD,

    url: 'http://example.org/download.php?action=download&id=1234',

    success: function(message, text, response) {

        var header = response.getResponseHeader('Content-Disposition');

        console.log(header);

    }

});

header variable will be something like attachment; filename=image.jpg. Now it's easy to extract filename part:

var filename = header.match(/filename=(.+)/)[1]; // image.jpg