I don't think there's a quick way to do this.
Since you know the numbers are odd, there can only be a maximum of 3 different numbers. For simplicity, I'm going to use 1,3,5 - but this will hold for any 4 odd integers with a range of 4 (because the distances from the mean will be the same). You can get the following 6 sets with corresponding means and deviations put in order (will clarify afterwards):
1 - {1,1,1,5}, mean 2.0, deviations (1,1,1,3)
2 - {1,1,5,5}, mean 3.0, deviations (2,2,2,2)
3 - {1,5,5,5}, mean 4.0, deviations (1,1,1,3)
4 - {1,1,3,5}, mean 2.5, deviations (0.5, 1.5, 1.5, 2.5)
5 - {1,3,3,5}, mean 3.0, deviations (0,0,2,2)
6 - {1,3,5,5}, mean 3.5, deviations (0.5, 1.5, 1.5, 2.5)
I only listed deviations instead of calculating standard deviation because it's not necessary to do all that work. All that work does is tranform the set of numbers into a single number, but it's easy enough to spot how many different sets there. Since there are 4 distinct sets of deviations (sets 1&3 and 4&6 share the same deviations), there are 4 possible standard deviations.
I haven't taken the GMAT yet, but this doesn't strike me as a type of question that would appear on the actual test. Maybe one of the experts can weigh in.
