When you say battery water, are you talking about the liquid that is inside of a lead-acid battery? That is a mixture of distilled water and battery acid (sulfuric acid, H2SO4)
Copper Hydroxide [ Cu(OH)2 ] reacts with diluted Sulfuric Acid [ H2SO4 ] to form water and Copper Sulfate [ CuSO4 ].
Edit: Though, why it's green I couldn't tell you...I'm not super good with chemistry, but this reaction is pretty much the way to create Blue Vitriol.