You could add the MONTH and YEAR functions or a CONVERT to group by.
SELECT RIGHT( CONVERT(char(11), FlightDate, 113), 8) Month,
COUNT (PaxNum) as TotalPassengersCount
FROM dbo.FlightDetails
WHERE FlightDate BETWEEN '1/1/2013' and '1/31/2013'
GROUP BY RIGHT( CONVERT(char(11), FlightDate, 113), 8);
PS. You can change the Month format