In a coconut shell… Yes.
There is in fact sugar in coconut water.
All coconuts contain sugar. How much they contain depends on the type of coconut and its age. Something to note though, even the coconuts with the higher levels of sugar still only contain around 2.95ml of sugar per 100ml, which is not a lot. (It's best to stay under 4.7ml of sugar per 100ml.) Of course, a bottle of coconut water — which is how most of us get our coconut water — is generally about 300ml. So. In one bottle there can be up to 9g of sugar, which is a little over 2 teaspoons.
But how much of that is fructose?
Well. Not so much. And this is what counts. A Brazilian study found the sugar content of an average baby coconut to be made up of: