They are probably unrelated - but possible related in a distant way. This is called a subconunctival hemorrhage and can be caused by coughing, sneezing, hard straining, and eye rubbing. Sometimes occurs for no apparent reason. Usually totally painless. Often, worsened by taking aspirin or other blood thinners. You should go ahead and see an ophthalmologist since you have some associated headaches. Ophthalmologists are medical doctors first (before eye specialty training) so they can often find clues from your medical history and findings to help in situations like this.
I am assuming that the vessels that you are describing are those located in the mucous membrane outside of the eye, covering the white part of the eye. Subconjunctival hemorrhages may result from trauma to the eye, bleeding disorders, heavy lifting. coughing, sneezing, vomiting, medications which reduce clotting time such as aspirin, coumadin and plavix and nonmedication herbs such as gingko, St. John’s wort, ginger, garlic, and diseases such as hypertension and diabetes.
Sandy T. Feldman, M.D., M.S.
ClearView Eye and Laser Medical Center
San Diego, California