The 5" patches will finish at 4.5", and since they're made into 3x3 squares, that's 3x4.5" = 13.5". Then that is cut once in each direction, so you lose 1/2" sewing them back together, so the squares are 13".
The next part of your question requires knowing how big you want you quilt to be. Full/queen is not a specific size. A full size bed is 59" x 78", if I remember right. I'd probably want the quilt to hang down at least 18" on each side, so it would work with a queen size bed, too. So, 59" + 18" + 18" = 95" and 78" + 18" = 96". Subtracting borders would be 80" wide by 81" long.
80" / 13" = 6.15 blocks. So, 6 x 6 or 6 x 7 blocks would work out. You might want a little more in length to fold under the pillows and cover them up. That's about 6" more in length needed.