Well there are 8 ounces in 1 cup so 1/4 of 8 ounces would be 2 ounces. There are two ounces in a quarter of a cup. There are eight ounces in a cup, therefore there are two ounces in a quarter of a cup.
Bear in mind that the US customary cup has 230,5882365 mL and the US legal cup has 240 mL; others "cups" are also known around the world. All these units are not from the metric system, creates confusion and it is strongly recommended to abandon them.