A fat quarter comes from dividing a half yard in two, split at the fold in the fabric. That gives you an 18 x 22" piece, instead of 9 x 44" piece, which may not work for what you want to do. Check the pattern to see if you need anything more than say, 8" in width...as most cuts at 1/4 yard are going to be off somewhere.
I'd get 3-3/4 yards (5/8 of a yard of the 6 colors needed). You may end up with little left over, but better than being a little short.
If you were to get 1/4 yard of 6 fabrics, you will probably have to do some piecing to make things work..and still may come up short.