[Mike795]

You could Takao-san mountain - it's only one hour by train from Shinjuku station on the Keio line (and the train stops right at the bottom of the mountain). It's right at the edge of Tokyo prefecture, but the mountains from there on are very steep, so haven't been developed and are just pristine nature. Except that Takao-san itself is a bit developed - there's a Shinto Shrine on top, and even a funicular railway to the summit - though it's also a nice and not very hard walk up too.

My second suggestion would be Kamakura. A very hilly town, with lots of Buddhist temples set amongst green, forested hills. It's also on the coast, and just along the coast is the picturesque island of Enoshima. To get there take the Yokosuka line from Tokyo station or the Shonan Shinjuku line from Shinjuku station. Alternatively the Odakyu line from Shinjuku goes directly to Enoshima.

Thirdly, if you like hot springs (bathing naked in the outdoors), you could head to Fuji Hakone national park - a bit further out - about and hour and a half on a different branch of the Odakyu line from Shinjuku Station. You can see Mount Fuji from there - but only on really clear days.

If you don't have time to make it out of Tokyo, there's always the Meiji Shrine, actually a big forest in central Tokyo, with the shrine at its centre, or Shinjuku Gyoen, a large national garden in Shinjuku, with a mix of formal gardens and relatively wild, wooded areas.

I hope that at least one of those options sounds good to you!