Hello, Welcome to Just Answer and thanks for your question. The small vessels in the nasal mucosa are very delicate and can burst from violent coughing. So can the vessels in the eyes resulting in subconjunctival hemorrhage. If it is a frequent occurrence, she needs to be examined by her medical provider. She needs that anyway since a constant cough is not normal. Bleeding can also be from a lack of blood platelets which a simple blood test would check.