From what I can see it seems that the fabric part of the zipper is the same color as the dress, and it is just the metallic (or plastic) zipper TEETH that are exposed. A sharpie/magic marker might not work to cover up metal.
Do check with your tailor on zipper replacement and show him/her a photo of the dress back. A more traditional zipper that is invisible will require that the back pieces of the dress have a little more width to cover the zipper and the dress may not be cut like that. If that won't work then the tailor would have to find a zipper with TEETH that match that dark green and that might not be easy to do.