Welcome to the forums. You're talking to other customers here.
I think you mean you have Dish for TV service. Dish doesn't use cable. It uses, um, a satellite dish.
Dish TV and Verizon are two completely different services. Neither depends on or cares about the other. The only potential conflict is if you plan to use in home coax to distribute both services. You can't put dish and Verizon on the same coax. Both services will work side by side, as long as they don't share any resources in your home.
Other than that, both companies will be happy to give you service and collect your money.