Unfortunately, the monitor you have cannot be H/W calibrated and I have no idea what it does with its internal LUTs. You have two options at this point:
- Return it for a calibratable monitor. My recommendation is the NEC P241W which I own two off (New they are $450 but I about them refurbished for $237) in addition to one of NEC's 30" model (LCD3090WQXi).
- Calibrate the display on each computer and that means finding a H/W Calibrator solution which works on both platforms. There is a good description of calibration (and miscalibration) for your specific display here. It even says that banding issues are minimal which is great considering that you have to calibrate at the graphics card level.
Never used that display but if it is calibratable as the technical specs suggest, then you only need to calibrate it once to the sRGB color space and make sure the color profiles of both machines specify sRGB.
To calibrate it you need a color-calibrator, I use NEC Spectra View but I suppose Spyder 3 (as already suggested by @dpollitt) will do as well. The only thing you have to make sure is that you are calibrating the display, not the graphics card. Otherwise you have to calibrate twice and will have banding and perhaps even inconsistencies, particularly considering you use HDMI on Windows.
Now, if you want to take advantage of non-sRGB colors, you can but it will be more work. Also, only color-managed application will show color colors. Everything else, including most web-browser (color management is supported by some modern browsers but there is no guaranty since files often come with the wrong profile) will show distorted colors.