Can falling unconscious on concrete cause spinal damage and/or a concussion?

Yes to both. At a minimum a head concussion with a variety of possible more complicated conditions, such as an intracranial bleed to a spinal fracture. Usually needs to be evaluated at a local hospital as circumstances that might have led to the loss of consciousness, if prior to fall, have to be medically cleared as well.

Yes, as depending how you land, can result in skull or vertebral fractures, and/or damage to brain and/ or spinal cord. As asked, the primary question is why you became unconscious, and of course, there are many reasons. If you have significant post-traumatic issues, these will need medical attention ASAP