When I read this I thought, "this must be a trick question."

Yes, deer will eat grasses of many types. I will see countless deer often when I drive, especially at night on relatively quiet country roads where there is fresh green grass, some clover and assorted other vegetation.
Of course, they eat all sort of other browse like young sappling shoots, acorns, apples, cedar bows, grainsand some pine tree ends (white pine) to name a few.

I have no idea why anyone wouldn't think an animal that browses on all sorts of other vegetation would eat tender green grasses.