A quick Googlebook search dug up this early mention of “quitting cold turkey” from 1827 (snipped only; featured above).
The Word Detective, writing in 2008, locates it in the 1920s, as an evolution from “talking cold turkey,” which meant talking without unnecessary ado. In other words, “cold turkey” is turkey without the bullsh*t.
Wikipedia entertains a few explanations, some of which relate to substance abuse treatment (because a withdrawing addict’s goose-bumped (ha!) skin looks like “a cold turkey carcass”), but that would seem unlikely given the 1827 use of the term.
A 2010 researcher, also using Googlebooks, turned up early 20th-century usages. “Cold turkey,” this researcher argues, referred to a wealthy man’s meal of choice–but he, too, locates the expression’s contemporary meaning in 1920.
Many terms radically change meaning over time, but it does seem unusual that the etymology of this phrase could involve the turkey (considered only as food product) in directly contradictory meanings (simple fare and a lavish dish).