Have only looked at the pattern and don't have one handy, but since each block finishes at 16", you'll need either 20 or 30 fat quarters, depending on how much of a border you want to do:
If you do 20 fat quarters, you'll have a center measuring 4 x 5 blocks or 64" x 80" plus whatever border you decided to get to queen size.
If you do 20 fat quarters, you'll have a center measuring 5 x 6 blocks or 80" x 96" plus whatever border you decided to get to queen size.
So your choice!