In Windows 7, Is there a way to snap a window to a quarter of the screen instead of half?

Not without 3rd party software, unfortunately.

You could try GridMove, which works quite well, and is open source.

You can just use Windows Task Manager, it's what I use at work with our Windows 7 machines.

Then Shift-select the four windows you want in your quad view and select either 'Tile Horizontally' or 'Tile Vertically' - both will produce the desired output.

You can use Winsplit Revolution.

In addition to having shortcut keys for positioning windows it also has a function called Drag'n'Go. By default you need to press Ctrl+Alt to enable it. When there are multiple possibilities you can use scroll button to switch between them.

Apparently program was discontinued and original website doesn't have downloads anymore. You can still download program from Techspot, there are installer and portable versions available.