You are right that usually conjunctivitis which causes pink eye will cause discharge. However, redness in the eyes
can be due to allergy, dry eyes
, a turned in eyelash, foreign body in the eye, corneal ulcer, conjunctivitis, cataract, glaucoma and in case of high blood pressure or coughing, it can be due to hemorrhage under the conjunctiva. It can also be a sign of stress, eye strain or poor eye sight. Please consult an eye specialist.
In all probably it is allergy related.