K.M.
In a word, yes. Allergies can definitely cause a cough due to the drainage tickling the back of the throat. My 3 year old and I both have been having coughing fits since things started blooming. Allergies also can start almost at birth. One sign of potential allergies in a baby is having eczema. My son had it as a newborn and has had other allergies since he was very small.
It's mostly just annoying and will go away on its own in a few weeks but you can try using a nasal saline solution that will help keep his nose clear and wash out some to the pollen/allergens. Or, you can call your pediatrician and ask about an antihistamine.