Can a disc bulges c-4 c5 and c-5 c-6 exciting nerve cause extreme consistent headaches ?

Unlikely: If your description is accurate, a disc bulge represents no significant issue, and is merely a coincidental finding. Even in cases of full-blown disc herniation, headaches would be fairly modest, in presence of far more neck symptoms. The etiology of your headaches lies elsewhere. See a neurologist and explore this further. Headaches can be controlled and even eliminated once cause uncovered.