There can be no equivalence.
A millimetre is a measure of length or distance in 1-dimensional space while a cup is a measure of volume in 3-dimensional space. The two measure different things and, according to basic principles of dimensional analysis, any attempt at conversion from one to the other is fundamentally flawed.
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Depends what size the cup is :/
Millimeters (mm) measure distance, while cups measure volume. The two are not compatible.
1 cup of water = 8 ounces, so 1/4 cup of water = 2 ounces
You would use millimeters for the measure of the capacity of a paper cup.
If you mean a cup as in "a cup of flour or sugar" you cannot convert cups to millimeters as a cup is a measure of volume were as a millimeter is a measurement of length. It cannot be converted.