What causes occasional nasal bleeding in the time of cold?

Irritated nose. Its likely that a person getting minor nosebleeds in the context of a cold has irritated their nasal passage with nose blowing and rough tissues. Going easy on this and using a saline nasal spray can help. Coughing up blood, bleeding that comes from the back of the nose, or that doesn't stop easily is more serious and should be evaluated by a physician.