Yes, you can!
It involves a lot of hacking up the computer. But the vga cable INPUT to the laptop monitor contains basically the same connection as a 15 pin monitor cable, as well as a couple extra pins (or they may be separate connectors in some laptops) that are used to power the display... you would have to have a spare 15 pin monitor cable (which you could cut off any old 15 pin spare junk monitor, but make sure it has all fifteen pins) and cut the cable off near the back of the old monitor. Then there is a LOT of testing to do to verify which pins lign up with the LDC's display. Of course your laptop would still need to be turned on to power up the LCD display, unless the power connector is contained in the DISPLAY connector, then you have opened up a new can of worms.
Basically, if you have an OLD dead laptop that you don't mind disassembling and spending a lot of time doing the trial and error reverse engineering task, then this would be a good starting point. The thing is, there is no guarantee that the Desktop will utilize 100% of the LCD's area, and since it has no function buttons, you will probably not be able to adjust the screen's vertical and horizontal positioning. It is a lot of work just to find out you can only see the top left hand corner of the desktop down in the bottom right of the LCD display, or even upside-down. I would recommend using a Volt-Ohm meter and plug the CUT OFF cable into an old PC, and turn it on, then test each wire's output, then with the laptop running unplug the LCD and test each one of the tiny connections where the LCD plugs into the laptop motherboard (not the LCD cable connector, but where it plugs into the mobo of the laptop itself.)
If it is a laptop that you need to have running tomorrow for work, I'd forget about it, since it really isn't worth the trouble, but IT IS possible to work out.
If I were you, I'd just create a Network Crossover Patch cable, and put WinVNC on both computers, then you'll be able to use EITHER computer backwards to display each PC's viewable desktop on the other.
I'm trying to find technical data on this myself, if anyone has ever done it successfully. I have a fairly old (800 mhz) HP laptop that someone lost the power cable, and the battery wont hold a charge, so they just dropped it off and said I could have it. I'd like to mount just the display on the wall and use it like a monitor, and I will succeed.
I will post results here F.Y.I. once I'm done but due to my sporadic schedule it may take a few days/weeks.