Asklemmy

That would depend on the way in which the individual became quadriplegic, any treatment they're receiving, and what parts of their body are affected by it.

It seems there's very cursory research showing some spinal injuries can increase your likelihood of developing conditions like pneumonia, and your risk of infection from most bacteria, but it doesn't seem to be true in all cases, nor has there been a lot of research as to if it persists forever, the exact mechanism by which it happens, or to what degree it can impair the immune system.

That likely isn't very relevant to the original question of asthma, though, unless the quadriplegic individual...

• Acquired any of a very small selection of respiratory viruses as a young child
• Received many antibiotics as a young child
• Became quadriplegic later in life and were exposed to a large quantity of non-pathogenic bacteria/viruses
• Exposed very little exposure early in life to non-pathogenic bacteria/viruses (e.g. from farms, pets, general non-sterile environments)

...since those are the primary mechanisms by which any form of immune reaction could be impacting the likelihood of asthma developing and/or getting worse/better.