This website might help:
http://www.joscountryjunction.com/a-...tumbler-quilt/
What I would do is first figure out the dimensions of the queen sized quilt you want to end up with. Then sew two tumblers together.
Measure the top edge after they are sewn and subtract 1/2" (for seam allowances). That will give you a width measurement for two tumblers. Say you want your quilt 100" wide and the two tumblers sewn together measure 6" wide (after subtracting the 1/2"). 100 divided by 6 = about 17 sets of 2 tumblers per row which = 34 tumblers per row. (33 tumblers might be closer to 100" wide.)
Measure the height of the tumblers and subtract 1/2" for seam allowances. I imagine this will be 3". If you want your quilt to be 120" long, then 120 divided by 3" means you will need 40 rows of tumblers. 40 rows times 34 tumblers per row would = about 1,360 tumblers.
Someone else may have more accurate measurements for you, but this is how I would approximate for myself.
Edit: You may want to decide before starting whether you want the finished quilt to have a zigzag edge like the the one at the website, or whether you want to simply cut off the side tumblers after you are finished piecing so you have straight sides to bind. Personally, I like things easy so I would do the latter.