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.

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.