It should work - even if you have the Quicksilver's original video card (the Geforce 2MX w/32MB VRAM), you should be able to use the monitor's large native resolution of 1920 x 1200.
However, if you plan to do more that surf the net and use Office (i.e games or iMovie), the little GeForce will struggle and I recommend that you plan on upgrading to a better video card. I recommend getting the $120 Radeon 9600 Pro 64MB from Other World Computing. It is a G5 card modded by OWC to work in G4 computers. It's cheaper than the slower Radeon 9000 and a little better than a Geforce 5200Ultra. It's also fully compatible with OSX Tiger's Core Image features. Alternately, you could shoot the wad and go for the $250 Radeon 9800 Pro Mac Edition 128MB, which will take anything you can throw at it.