As a 22 year-old, can surgery be done to correct a cleft lip palate nose deformity?

Yes. Discuss this with a plastic surgeon who specializes in this management.
Correct facial cleft. Indeed, corrective surgery for cleft lip and/or palate can be done at any age. In fact, adult or adolescent cleft surgery often results in a better esthetic and functional outcome because the structures involved have not been disturbed by previos surgery and growth and development is unimpeded by scars from surgeries done as an infant or younger child.
Yes! Revision surgery to improve a cleft lip nasal deformity can be done at any age. I recommend going to a major university with a cleft lip and palate surgical team.