Wordorigins.org

This phrase meaning suddenly, without preparation or to speak frankly is originally a reference to food. Cold turkey is something that can be prepared quickly and with little effort.

The earliest known usage is from 1910 in Robert Service’s Trail of ‘98:

I’d lost five thousand dollars..."cold turkey.”

The sense meaning to speak frankly dates to at least 1920 in a citation from T.A. Dorgan:

Now tell me on the square—can I get by with this for the wedding—don’t string me—tell me cold turkey.

The sense meaning to quit an addictive substance suddenly is from at least 1921, when it appears in the Daily Colonist of Victoria, British Columbia on 13 October:

Perhaps the most pitiful figures who have appeared before Dr. Carleton Simon...are those who voluntarily surrender themselves. When they go before him, they are given what is called the “cold turkey” treatment.

There is an explanation that the pasty, goose bumped skin of an addict going through withdrawal resembles cold turkey skin and this gave rise to the term. But this is not borne out by the fact that the addiction sense is a later one.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang)