The first Friday of a month is in the range from 1 to 7; therefore the third Friday is from 15 to 22. Now you simply loop from the 15th to the 22nd of the given month and stop when that date is a Friday.
The 10th business day works along similar lines:
Date determine10BusinessDay( int year, int month )
var counter = 0;
var day = 0;
while( counter < 10 )
if( isBusinessDay( Date.of( year, month, ++day ) ) )
return Date.of( year, month, day );
Of course you need a proper implementation of the method isBusinessDay(). Best would be to use a BusinessCalender as the base for that implementation, but a simple check whether the given date is a Sunday would do also, depending on your requirements.