My son had allergy testing at about six months because of persistent asthma. The allergist told me seasonal allergies are unlikely (note he didn't say impossible) because a child has to be exposed to the allergen for a while before symptoms begin, and that dust, mold, cockroaches and other environmental allergies that aren't seasonal are more common in very young babies.
In our case it seemed that DS's symptoms were not allergy related at that age, but a year later he did develop seasonal allergies that he still has.