To figure out how many quarter cups make two thirds of a cup we need to first find a common denominator, which is 12.
1/4 = 3/12
2/3 = 8/12
Now we can do some very simple math dividing 3 into 8:
8 / 3 = 2.66
So it would take 2.66 quarter cups to equal two thirds of a cup (just a bit more than two and a half quarter cups).
If you're baking and need very precise measurements, but don't have a 2/3 cup measure, it might be better to use spoons: 2/3 of a cup is 10 tablespoons, and 2 teaspoons.
Find lots of tips on making measurement conversions at EgglessCooking.