Green’s Dictionary of Slang

conk v.3

also konk
[conk n.2 (2)]

(US black) to straighten hair with a mixture based on Congolene.

[US]Chicago Defender 21 Oct. 12/7: Others conk their hair down and it gleams as brilliantly as if they were born in India and it looks fine.
[US]N. Heard Howard Street 149: He looked upon hair-conking with scorn.
[US]E. Torres Carlito’s Way 9: You had a pimp name of Red Conk on account of he conked his hair red.
[US]Eble Sl. and Sociability 82: Many terms that originated among African Americans remain for the most part in that community, like conk ‘to straighten hair’.