Thank you for your query.
In allergies one can have nose bleeds but that only if the allergic rhinitis
is very severe. In your son's case it is a very unlikely cause. As he is also complaining of headache has his blood pressure been recorded? Hypertension
can explain both the headache and the nose bleed.
I hope I have answered your query to your satisfaction.
Wishing your son an early recovery.
Rajiv K Khandelwal