I am going to test my 6000 watt generator. It is recommended that I ground it to a copper rod driven into the ground. I alread have a copper rod that is used to ground my telephone and cable lines. Can I use a length of coax cable to run from my generator ground to the copper rod?

I'm all for reusing existing materials when you can, but I'd strongly recommend against this particular idea. Coax is not designed to handle the amount of current necessary to safely keep the generator grounded in case of a malfunction. It also may not be copper, in which case it could corrode where it connects to the generator or ground rod.

Bare copper ground wire (usually #6 or #8 for that size generator I think) is relatively cheap (probably something like 60-80 cents per foot). If it was me, that's what I'd use. And if the generator had to be more than about 20 feet from the existing ground rod, I'd just pay the $15 or so it would cost to buy a new copper-clad ground rod, a clamp, and a few feet of #6 wire; and I'd pound the rod into the ground right next to where the generator's going to be.

I am sure the instruction manual for the generator will require at least a min of #8 copper wire for the ground.