UPDATED FOR TAX YEAR 2019
No - The earnings generated in a 529 plan are not subject to federal or state income taxes while the funds stay in the account. Contributions to a 529 plan are made with after-tax dollars. Earnings within a 529 college savings plan occur on a tax-deferred basis.
When the money is used for qualified education expenses, the distributions from the 529 plan are not subject to federal or state income taxes.
[Edited | 4/6/2020 | 11:58am PDT]