In your old machine, they all pointed to G:\Library, right?
So start out the same way in your new machine. The targets are still in G:\Library, and the .lnk files, wherever they are, still point to G:\Library.
Make \NewLibrary on drive Q:. Now just right drag&drop everything in G:\Library to Q:\NewLibrary, and select "move". Or move the whole Library folder to Q:, and then rename it NewLibrary.
In either method, the .lnk files will update themselves, no? (as long as you *move* the files, not copy them and delete the originals.)