Absolutely it can give you a headache
Yes, I have gotten headaches from Wi-Fi also. One person writes "Wifi can not affect your body in any way," and many people point out that if you got a headache from your PS4 then you would get one everwhere because there are lots of wi-fi signals everywhere.
I would like to refute both of these statements.
"Wifi can not affect your body in any way." This is not true. Wi-fi exists, it is physically real in the same space as your body, without it being physically real there is no way it could carry information. It is a form of light. All electromagnetism (light and electricity) is tied to all other electromagnetism (light and electricity) — and all matter, according to the laws of electromagnetic interaction, Maxwell's equations, and more precisely the electroweak interaction (which is even more fundamental, though I do not understand it).
Electrical signals in your nervous system (the neurons firing in your brain) and charged ions in your body are electrical signals as well, and they are subtly affected by the ambient electromagnetic fields. Just like microwaves from your microwave oven, gamma rays and ultraviolet from the sun, visible light from your desk lamp, infrared from your fireplace that warms anyone exposed to the flames, and TV and radio signals. They are all the same elementary particle (the photon) resonating at different frequencies. Photons are waves, just like sound... As in the case of an opera singer who can shatter a crystal glass with her voice when she sings at a particular pitch (the same pitch the crystal makes when you strike it with your fingernail) certain frequencies will affect your body (resonate) more strongly than others will. Wi-fi is at a wavelength that does not affect humans in a strong way, however, just like supersonic audio frequency which a dog can hear but a human cannot, when the wi-fi gets bright enough it can still affect your body.
People have made a valid point that we are in a bath of wi-fi all the time, but that does not mean that those connections are transmitting at a high rate of bandwidth right next to your head. Intensity of the EM radiation decreases as a square of the distance from the source. So, if you are 20 feet away from a signal, you are getting 4 times the exposure than you would get at 40 feet. And if you are 10 feet away, it's 4 times the exposure you got at 20 feet. The geometry is more complex when you have not one, but two resonating antennas (your wireless router and your ps4), but it still will fall off at roughly the square of the distance. Wi-fi does not need a very strong (bright) signal in order to detect a network, so when you see the dozens of wi-fi connections active in an apartment building it is important to remember that they are very dim by comparison to the PS4 right next to you.
Another thing to consider is that just because a wi-fi network is active does not mean that it's actively transmitting at a high rate of bandwidth. Wi-fi can carry up to, say, 20 megabits (20 million ones or zeros) every second, but usually when a signal is active and the user is just browsing the web, then that is using, on average, perhaps one hundredth of the total capacity of the wi-fi network. This is equivalent to a dim flicker. On the other hand, if a user is streaming video, playing a game that requires high bandwidth, or running a file transfer, then the "brightness" of the signal increases a huge amount. This is equivalent to a bright and steady light. It's not just an analogy - Wi-Fi, and cellular transmission are both actually, literally a color of light that we can not see. I found an interesting science article on the subject by searching Google for "wi-fi wavelength"
I'm not proposing that wi-fi will give you cancer - the wavelength needs to be at a resonance frequency with your DNA to cause mutations and give cancer, and wi-fi is not anywhere near that frequency. I'm not even suggesting that it is necessarily harmful. But I am refuting the myth that it can not affect your body. All EM affects your body, it's just a matter of how much. And I'm also pointing out that your video-streaming ps4 might literally be thousands of times as "bright" right next to your head as someone else's idle wi-fi network is when it's flickering down the hallway.
Some people may have a high degree of tolerance for extremely intense (bright) wi-fi signals, and the way it affects their body will never, ever manifest as pain. Other people who are more sensitive might get a headache when they are very close to a strong signal that is in use at a high bandwidth. It's not necessarily the source of your headaches, and the placebo effect becomes every bit as real as a wi-fi signal once it manifests in your body... but I absolutely wouldn't discount it as impossible or even all that improbable that wi-fi can contribute to headaches.
Thanks for considering my opinions.
p.s. it is not likely to be a true "migraine" unless you also experience some degree of "shimmering" blindness prior to the onset of pain and nausea.