The word conk is slang for nose.
From Oxford English Dictionary (OED):
conk, n.1
The nose.
But conked also means to punch on the nose. From OED:
A punch on the nose or head.
1821 P. Egan Boxiana III. 338 Spring however conked his opponent,
I think it’s therefore fair to assume that conked out originates from someone being punched on the nose and knocked out.
But that is purely speculative.