Can birth control pills cause subconjunctival hemorrhage?

Generally not. Medications can affect people in different ways. I'm not aware of this being a common side affect.
Occasionally. In some women, birth control pills cause some degree of a bleeding tendency. If the eyes are predisposed to have a subconjunctival hemorrhage, this can be worsened by the use of such pills. But they do not cause the initial bleeding which requires fragile vessels and some degree of trauma.