I take it you are talking more about nutritional nitrogen than composting nitrogen since the two are actually different in their action.
Farmers are most interested in protein content, they are after all trying to feed living organisms.
fresh greens will generate heat, this is because of the internal moisture content and the nitrogen beginning to break down into nitrates and nitrites (exothermic reactions),
this doesn't occur in dried hay materials since the breakdown isn't going to occur from moisture loss. What you get from dried materials getting wet is mold growth.
I have never gotten a
compost heap to heat from using hay for a green and I've been composting since 1964.
Here is a little experiment for you to try for yourself. Fill a black garbage bag with nice, dry hay and seal it with a twist tie. Fill a black garbage bag with fresh cut, green grass and seal it with a twist tie.
Place both bags in the same area, wait two or three days and place your hand on the outside of each bag (you can insert a long probe thermometer through the bags if you want to record an actual temperature).
Write down your findings, make decision from your notes.
Items that should be considered browns will not heat of their own accord, items that should be considered greens will.