There's a few things you can try to narrow down where the problem might lie. One thing you don't mention is if the green line is on the actual image, or only shows up on the LCD monitor.
If the line is in the actual image after transferring it to your computer, try formatting the memory card in the camera, or try a different card to see if the problem persists. If it only shows on the LCD, count yourself lucky because you can continue to use your camera, albeit with an annoyance to deal with.
However, it sounds to me as though you've got either a dead or hot pixel on the CCD which will cause problems in the adjacent pixels creating a "line". Take a look at
this page and
this page. Unfortunately, you will most likely need to send it in to Nikon for repairs.
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