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.