Well it doesn't happen to me OR Mrs sqad, but it is a well known syndrome.
The cause is not known, but the theories vary from, coming into a cold atmosphere from a warm environment thud irritating the sinuses, to a direct stimulation of a ganglion (nerve centre) in the brain similar to trigeminal neuralgia.
I get really dry eyes and aircon tends to dry them out even more, I get headaches when they are really dry though it tends to affect my vision too when really bad.
I had a confusing situation of a number of pretty different eye prescriptions and couldn't understand why until one optician told me that I should always put eyedrops in before having my prescription checked as when my eyes are dry the prescription changes, hence the different results.
I'm also big on fresh too, aircon can feel quite stuffy to me if there is no fresh air.
Another thought, does she wear contact lenses? I was getting awful headaches and had to stop wearing mine as my eyes are so dry they suck the moisture out and the lenses onto my eyes, aircon can be quite a dry environment.
Yes air conditioning can cause headaches because of following reasons..
1. Dehydration.
2. The cold causing muscle contraction.
3. The cold causing blood vessel contraction.
4. The air conditioner may be circulating stale air, chemicals, or other allergens from a ventilation system that hasn't been cleaned out properly.
5. The noise the air conditioner is producing may be sending out a frequency you are unable to handle.