You guessed right, there is a white balance calibration option in the service menu, this might fixed your green tinge.
First steps
Video settings on Xbox
switch to 1080p, 10 bit colour and PC full range.
If you want to skew this calibration towards HDR content check the YUV 4:2:2
In the youtube app on the Xbox run this video.
https://www.youtube.com/watch?v=_J_rR5qWN30
go into the service menu and go down to ADC/WB
go to the ADC Result screen and take a photo of these settings as backup
look at the white balance screen and take a photo of this screen as backup.
When you've got a note of those screens, ensuring that the youtube video is runnin in the backgrounf
You don't want ANYTHING else on screen, no play buttons, pause buttons etc.
go to HDMI calibration and chose left or right. All bring well you'll see the screen changin a little and eventually the service menu will change from wait to success (I had to do it a couple of times as it failed)
Of course because entering the service menu changes all of the picture settings, you'll need to set everything up again, but hopefully your green tinge will be gone.