the phrase is "quit cold turkey" (no 'the') It means to quit something that has become a habit for you without diminishing your consumption first. It is often used in the context of quitting cigarettes, drugs and\or alcohol, meaning that the person went from using it to excess to not using the substance at all. It can also be used to describe other things that are "addictive", but more in jest, such as "I was getting fat, so I quit eating hamburgers cold turkey".

It means you stop using something suddenly and completely, rather than gradually reducing your use over a long time. Drug addicts, for example.