I don't have this experience before, but my friend has. Usually the eye drops will not cause you this. The eye drops are used to get rid o the invisible bacterium in your eyes and make your eyes become moisture. When you get the headache, the eye drops may stimulate your eye nerves. You may be sensitive to this type of the eye drops. You should change another type of the eye drops.
Yes. Some eye drops may cause headaches if taken internally. In fact, most of eye drops contain visine. So, though eye drops can help the eyes sometimes, it also has side effects to our health. It can reduce bloodflow and oxygenation to the surface of the eye if you often use it. For long-term side effects, some people even experience a rapid heartbeat.
Ok, looks like you have got some troubles with your eye drops. Anyway, we will not have headaches with the use of eye drops. However, according to some cases, some people are sensitive to such drugs and those drugs would inevitably generate some negative impacts on your eyes and health, thus, some problems would emerge. Anyway, try to take care of your eyes and try to resort to other means of protecting your eyes and vision.