This could be one if not a combination of issues: For the LCD display to work, we need to check the graphics card, driver, proper screen working and cable connecting screen to system board.
You would need to systematically isolate the issue, if you've got the proper driver, it won't be the issue.
Plug an external monitor in to see if the same green line appears. If it does, then it's not just the LCD screen. If it only appears on the LCD, then it's either the screen or the cable that connects it to the board. You can try applying a steady pressure to the line it see if green disappears, the monitor may not be the issue.
You may have to check to see if the cable between screen and systemboard is kinked, damaged or loosened. This requires you to possibly take apart the laptop.
This is just my opinion, other's may have more insight in this than I do, I hope they chime in!