Redness in the eyes can be due to allergy, dry eyes, a turned in eyelash, foreign body in the eye, corneal ulcer, conjunctivitis,
glaucoma and in case of high blood pressure or coughing, it can be due to hemorrhage under the conjunctiva.
Hope this helps. It is difficult to comment beyond this without examining. Do consult an eye specialist. Please let me know if there is any thing else and do keep me posted. Take care!