To directly get the counts for each group:
setkey(test, Month, Complaint)
# may need to also add allow.cartesian, depending on actual data
test[CJ(Month, Complaint, unique = TRUE), .N, by = .EACHI]
# Month Complaint N
# 1: 2013-10 A 0
# 2: 2013-10 B 0
# 3: 2013-10 C 5
# 4: 2013-10 D 4
# 5: 2013-10 E 0
# 6: 2013-11 A 1
# 7: 2013-11 B 0
# 8: 2013-11 C 4
# 9: 2013-11 D 0
#10: 2013-11 E 0
#11: 2013-12 A 1
#12: 2013-12 B 0
#13: 2013-12 C 0
#14: 2013-12 D 2
#15: 2013-12 E 0
#16: 2014-01 A 0
#17: 2014-01 B 0
#18: 2014-01 C 1
#19: 2014-01 D 0
#20: 2014-01 E 1
#21: 2014-02 A 2
#22: 2014-02 B 0
#23: 2014-02 C 6
#24: 2014-02 D 2
#25: 2014-02 E 0
#26: 2014-03 A 1
#27: 2014-03 B 2
#28: 2014-03 C 3
#29: 2014-03 D 0
#30: 2014-03 E 0
# Month Complaint N
See first revision of the answer if you want to have the counts in the full data.table
instead of summarizing.