Yes, the headache will cause the broken blood vessel in eyes. However, it will not last long enough if you feel good and not feel pain in the head. It is temporary. You should have a good rest for the eyes and not use them in front of computers or tv for a long time. You should not eat spicy food which may stimulate the blood vessels.
Yes, headache can cause broken blood vessel in eyes. Headache can be divided into two sorts: one is the common cephalagra without omen, another is the cephalagra accompanied by omens. The cephalagra with omens can be divided into two periods: the foreboding period and the headache period. In the foreboding period, the symptoms of visual sense are common such as photophoby, flash of light before eyes, eye delusion, afterwards appear visual field defect, dark spot, partial blindness or temporary blindness. The headache manifests after the omens begin to diminish. Pain begins in an orbit or back of orbit, then it spreads to the head and neck. With the degree of headache increases, some manifest flushing face with a large number of sweat and conjunctiva hyperemia while some manifest pale face with drooping spirit. Usually, after taking a good sleep, headache is relieved and bloodshot eyes gone.
It is hard to tell whether headache will directly leads to broken blood vessel in eyes. I think you should find out the reason why you got headache. For example, if you have a headache because of the head trauma, the injury in the face, will cause broken blood vessel in eyes. If your headache is caused by high blood pressure, it might cause broken blood vessel in eyes too. If you do have a headache and the subconjunctival hemorrhage, please go to the hospital for professional help.