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 ) ) ) 
        {
            ++counter;
        }
    }
    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.