A veneer is a manmade finish. If you are sure that your granite is the real thing then proceed with the following directions. If you think it might be a different stone like marble then skip the acid cleaning. The only acid that will etch polished granite is hydrofluoric acid which is not present in the products I described.
In order cover the grout effectively you will need at least 1/8" clearance for the coating of grout you will use.
Scrape out enough grout to meet this requirement from the high spots in the joints and afterwards clean all stone and grout with a acid based tile cleaner and a stiff plastic scrub brush. A solution of CLR (lime remover) will also suffice. Allow the fireplace to dry overnight.
Instead of using mortar, you will need a gray (or whatever color you choose) non-modified grout with no polymer or latex additives. You will also need (1) gallon of acrylic latex admixture. It's a milky white fluid and easily found at any hardware store. This will be the what bonds the new grout to the old grout. Mix the grout with this liquid until it becomes a thick sticky paste. Do not add water.
If your joints are an 1/8" or smaller purchase a non-sanded grout otherwise use sanded grout. Grout a small area at a time and wipe clean with a good thick tile sponge. Mix only enough grout that you can use in 20-30 minutes. Acrylic latex admixture will bond great to cement but the downside is it will try to bond to the stone. Remove any of the residue quickly with clean water. If your stones are flat use a grout float to grout otherwise I think it would be easier to use a margin trowel to push the grout into the joints.