How many grams is a quarter cup of grated mozzarella?

A quarter cup of grated mozzarella (shredded mozzarella) weighs

28g

That's 1oz. In a 250ml metric cup it's 30g.

This uses the US cup, the size American recipes mean. A quarter cup is 59ml by volume.

Other amounts of grated mozzarella

1 cup: 112g
Half a cup: 56g
1 tablespoon: 7g
Full table for grated mozzarella

Where these numbers come from

USDA FoodData Central, SR Legacy, food 01026: “Cheese, mozzarella, whole milk”. USDA lists these household measures for it:

1 cup, shredded = 112g
1 oz = 28.35g
6 slices = 170g

Fractions of a cup are the cup weight scaled down. The tablespoon figure is the cup weight divided by 16, since USDA doesn't list a tablespoon for this ingredient. The teaspoon is a third of the tablespoon. Metric-cup figures assume a 250ml cup, and Australian tablespoons hold 20ml. Weights under 10g are rounded to 0.1g, everything else to the nearest gram. Full source list.