Would a surgical puncture to the skull relieve a pressure headache?

Am doctor. Outside of very rare and specific causes of headache, no this wouldn't fix anything, just put you at risk for infections.

sighs Oh god. No, it wouldn't. Here we go.

The name pressure headaches come from the feeling of pressure,not from actual pressure - actually most pressure headaches are not caused by a process within the cerebral cavity at all. Most are related to sinuses, especially the frontal sinuses, teeth, the middle ear/auris media, tension headaches(which are usually cervical/muscular). Only migraines as a cause for pressure headaches are actually within the cerebral cavity but have nothing to do with actual pressure. (Okay, actually we don't really know how they start - but we are really sure it's not the pressure)

As the brain matter itself has no pain receptors it does not actually feel pain - but the root of nerves like the Trigeminus, Occipitalis and the layers of tissue covering the brain (Meninges) as well as the vessels can absolutely feel pain. A lot of pain.

Cerebral bleeds, post aneurysm or trauma, usually happen in a very acute form or very slowly - with patients suffering from minutes to month from occasional headaches,often rather mild and not "pressure" headaches. Often after an initial knockout punch. (And at that point I must remind people that someone who has been knocked out needs a brain scan urgently. It's NOT something to joke about and feel ultra manly,eh, toxic masculinity, about)

The acute form is totally different though - it's one of the nastiest ways to go. Patients have a sudden onset of massive,life-changing pain, described by survivors as a catastrophic, sharp(*), pain associated with a fast onset of other neurological symptoms.

Another misconception is the fact that the cerebral cavity has no openings - it has quite a few and when the pressure within the skull is increased by a pathologic event like a cerebral bleed or the swelling of the brain the brain gets pushed in these holes - mainly the hole connecting to the spinal column. Which is why cerebral bleeds are so deadly - because the brain goes "stem first" through there - and it's too small even for the brain stem. And the stem is what regulates breathing,etc.
Other nerves get simply sheered off/cut by the pressure. (This is part of the pain mentioned above - Large nerves being cut by your own blood pressure)

Nevertheless even in these cases the removal of large parts of the skull as it is done by neurosurgeons does not alleviate the pain - the damage to tissue, nerves, etc. is done. The goal is to give the brain room and stop the bleeding. But thankfully we have good options for pain relief these days and these patients are in a very deep sedation/coma for a good reason.

(There are of course rare special cases like hydrocephalus cases, but that's a bit far fetched here)

So no, it would not help a bit painwise ,even in the most extreme situations.

So as a healthcare professional I generally would advise against drilling into one's head. It will make a lot my colleagues very angry.

And if you really feel you must do that please consult the nearest psychiatrist first. And if you then still feel like you must do that, make sure to do it during business hours, make also sure you livestream a video of it for the loolz of a lot of HCPs around the world, get a testimony recorded so your next of kin can sue the earlier mentioned shrink that he did not admit you to a closed ward and most importantly get a organ donor registration.

And due to a call a decade back I might also add: DO NOT drill into your own Sinuses as well,even if they are clogged up. Even if you are a doctor. You might accidentally (almost) lobotomize yourself.

Source: Am a paramedic. Seen shit. Seen people do shit that started as an internet hoax.

Do not put a hole in your skull. No. The headache is not caused by cranial pressure.

Your sinuses may be inflamed, and relieving that pressure might provide temporary relief (like pressing on your forehead) but a hole in your skull wouldn't fix the underlying sinus problem, and would likely result in worse infections and irritation.