Battery "water" is sulfuric acid (H2SO4).
It turns green when it reacts copper hydroxide because copper (II) ions are blue in solution.
From the reaction that should occur:
Cu(OH)2 (s) + H2SO4 (aq) = CuSO4 (aq) + H2O (l)
Copper (II) sulfate should be produced. In solution, this is a deep blue color. The fact that it is green as opposed to blue means that there is something else in your H2SO4 that the copper is forming a complex with. For instance, in citric acid, copper (II) ions form a light green complex. So, you've got H2SO4 with some sort of contamination. I'm not sure what it could be.