You cannot.
Milligrams per centimetre square is a measure of planar density (mass per unit area), whereas parts per million is a pure number, without units. The two measure different things and, according to the basic rules of dimensional analysis, conversion from one to the other is not valid.
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As squared meters are two-dimensional and cm are one- dimensional it is not possible. If you meant cm squared then: 1 Meter squared = 10000 Centimeter squared. Using this you can use simple multiplication to convert.
1 square centimeter equals 1 centimeter squared
Since a centimeter is 10 millimeters, it follows that a square centimeter is equal to 102 = 100 square millimeters.
1 square centimeter = 0.0001 square meters
You either divide the giganewton by one million or one thousand. I think it's one million. so it would be a millionth of a newton, or 0.000001 N.