The most common way to do this is to work with JOGL, which is a thin layer of Java on top of the OpenGL package. OpenGL supports stereoscopic 3D when paired with related equipment. You will need to use the right equipment, that is, a graphics card, since not everyone will support stereoscopy.
Here are some links to help you get started with JOGL and stereoscopy:
Keep in mind that JOGL is a very low level package and it’s not easy to learn or work with. Of course, you cannot “convert” your Java application to 3D with it, unless it was written in JOGL for starters.
In the interest of comprehensive answers, it is also possible to do this in Java3D, which is a simpler, easier to use package than JOGL.
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