No. Producing green mana doesn't make a land a Forest. Having the subtype Forest makes it a Forest. This includes not only basic Forests, but cards such as Stomping Ground, Dryad Arbor, Sapseep Forest, and any other that has "Forest" in the type line.
All Forests can produce G, but not all lands that can produce G are Forests. Some are just printed with the ability to do so. Only lands with Forest in their type line are Forests. This includes Forest obviously, but also cards like Breeding Pool or Murmuring Bosk.
All Forests can produce G, but not all lands that can produce G are Forests. Some are just printed with the ability to do so. Only lands with Forest in their type line are Forests. This includes Forest obviously, but also cards like Breeding Pool or Murmuring Bosk.