Normally, we say "I have a headache." We would say "This is making my head hurt" facetiously to indicate some complicated situation or problem is driving you crazy (frustrating you). Or "Ow. After looking at this mess, my head hurts."

Mojave, thank you very much!

Both are fine. "I've got a headache" is more specific. "My heat hurts" can also mean you have a headache, but it could be more general. Maybe you cut your head or you have a bruise, etc.