Is there a relationship between bruising and allergies?

No: Common nose or eye allergies don't cause bruising. Bruises are from blood leaking out of capillaries (tiny blood vessels) in the skin. Bruises can mean there was trauma to the area, the blood vessels are abnormally fragile and break easily, there are too few platelets to help blood clot, the blood is "thin" and not clotting fast enough, etc...