Kent, 64, has worked as chief executive of the beverage company since 2008 and will continue as chairman.
He said it was "the most wonderful and unique privilege" to work for the company, and will officially step down 1 May 2017.
Quincey, 51, has been with the company since 1996 and was called a prime example of Coca-Cola's "talent pipeline in action" by Kent. He has been in his current position since August 2015.
As chief operating officer, Quincey worked to expand product offerings, introduce smaller packaging sizes and reduce added sugar in Coca-Cola products.
Read more: Coca-Cola is ditching Coke Zero
I am committed to continuing my strong partnership with Muhtar, our talented management team and associates, and our valued bottling partners to continue this momentum and capture the enormous opportunities in front of us.
Sam Nunn, independent lead director of the company's board, said Quincey "has the ideal combination of skills, ability and experience" for the job.
Warren Buffett, chairman and chief executive officer of Bershire Hathaway, the company's largest shareholder, said the company had made a "smart investment" in its future.
The company's shares went up more than 2 per cent in early trading.