I would color correct on the better monitor and not try and match with the secondary that is "close".
The difference can be slightly off program to program, but first make sure both files are in RBB, same Bit Mode (8 or 16) most likely 8. Decide what to adjust with, color balance or HSV, HSL and make notes. So after you make the adjustment you'll know the steps. Or a preset per adjustment layer is helpful too when making the adjustment through Photoshop.
Hope that helps a little.
I worked in a studio wher the IT techs came around weekly to do monitor calibrations. We all had two monitors with one assigned as the final color check. Being that the second could never match. (different brand, model or year).