I have this recurring cough, that comes about this time of the year and lasts months. I show no other symptoms. Should I be worried?

Allergy... Your seasonal cough is most likely bronchospasm related to seasonal allergies. You may need a bronchodilator and perhaps a steroid inhaler to control the cough. See your doctor for evaluation and treatment.
Cough every spring. It seems you have spring time allergies and it comes at the same time pollen count is high, you do not have to be worried.If you want, you can try over the counter allergy meds like claritin, (loratadine) Allegra or zyrtec.It could be mild case of asthma which goes away without treatment.You can discuss with your doctor or see an allergist if you want to be sure.