I doubt it - the VGA -> DVI adapters I've seen are for connecting a VGA display to a DVI-I port on a computer. If you take a look at the DVI plug you'll see that it likely won't have anywhere near the same number of pins as a normal DVI connector. Furthermore, it will likely have no electronics in it to convert between VGA and DVI - the adapter works by forcing the video card to run the port in VGA mode. I could be wrong, but I have never, ever seen an adapter that actually converts VGA signals into DVI.

Your best bet, if you can't get an old Mac monitor, would be to get a DA15 -> VGA adapter, and an old 15" VGA LCD, and use that.