I'm want to emphasise date period(2 dates: start and end) in table cell. Usually it's green for start and red for the end, but I believe it'll be too distracting...
The solution could be to use softer/lighter versions of colors. In your case, I would choose
#BBFFBB for green and
#FFBBBB for red. I used those exact colors on a couple of my websites. They are much easier on the eyes and also fit nicely on any modern-looking application. Alternatively, you can use Color Schemer to choose appropriate colors. In your case, choose a plain red/green color and then press the button Lighten Scheme until you get the wanted result. Good luck!
Is it just one instance (row) you want to emphasize out of many? If that's the case, perhaps just a text weight change (bold) could help, plus a light background to the cells in the table. Or even highlight the whole row itself.
If it's more than one row, then color could be a distracting element. Also, using red may clash with other elements designed to show errors/alerts. Are the dates being emphasized because the user needs to take action on these items?
If you have a mockup, it would be great to have a look.