Is it possible to ferment fresh coconut or its water?
In a word, yes.

According to MakeSauerkraut.com (of all places)


  To make fermented coconut water, the sweet water is mixed with a
  culture (milk kefir grains, water kefir grains, whey from yogurt or a
  powdered starter) and left to ferment for 18 to 48 hours in a warm
  place.


There seems to be a lot of recipes out there for such a thing, a quick internet search for "ferment coconut water" should yield more recipes, methods and the like.