In a pond without fish, there are less factors influencing the Nitrogen cycle than would be if fish were present, but it is still functioning. All organic matter, when it decomposes, gives off Ammonia as a by-product. Some more than others. The nitrifying bacteria transform this ultimately to Nitrate which is used by algae to fix (eat) Carbon. Algae also have the ability to take up Ammonia directly by assimilation where it is internally converted to Nitrate.
If you are getting green water in a Water Garden (no fish) then it is because of an elevated presence of Ammonia and/or Nitrate. If it were string algae then it would be Ammonia and/or Phosphorus. In either case, you have decaying organic matter.