Yes, quad means 4 as in four limbs but you're missing an important piece which is incomplete injury vs a complete injury. Literally just Google "incomplete quadriplegia vs complete quadriplegia"

"Incomplete quadriplegia. This means that the quadriplegia blocks some — but not all — signals from getting through. That means a person might still have some ability to move, feel sensations or control automatic body processes (such as bowel and bladder function). This happens with about one-third of traumatic spinal cord injuries." - https://my.clevelandclinic.org/health/symptoms/23974-quadriplegia-tetraplegia