Just to confirm, everything is fine when you stop using the new monitor? so the problem appears to just be the monitor?
If that is the case, I would suspect LCD inversion to be a potential cause of your problem. You might read about it here http://www.techmind.org/lcd/
In short, an LCD pixel's voltage has to invert every frame to prevent degradation. This creates a flickering effect. All LCD panels will do this, even if they are marketed as flicker free. The way inversion is implemented and perceived will vary between monitors and LCD technologies. However, a 240hz monitor will flicker through inversion at 4x the frequency of a 60hz monitor. My suspicion is that this higher frequency flicker is giving you a migraine.
Temporal dithering is another potential cause, your GPU will alter its output every frame even if displaying a static image to improve colour quality. Similarly to inversion, using a 240hz monitor will dither 4x faster than a 60hz, potentially making migraines more likely.
If possible, setting your 240hz monitor to a 60hz setting will resolve any problems caused by faster dithering. It might solve the inversion problem, but I am not sure if setting it to a lower framerate will actually change how inversion is implemented - it may continue inverting at 240hz regardless of the framerate you set it to.
Further assuming you have a migraine, it may take you several days to feel 100% again.