How do I enter state income tax paid with quarterly estimate payments for 2016 taxes

You enter state estimated payments under the federal side (because they can be a federal itemized deduction)

You can type estimates paid in the search box at the top of your return and click Find and it will give you a link to Jump To it.

OR
To enter State Estimated Taxes Paid (including a state estimated payment made in Jan 2016 for 2015),  you have to first enter it under the Federal side,  go to
Federal Taxes (or Personal for H&B/Self Employed version)
Deductions and Credits
Then scroll way down to Estimates and Other Taxes Paid
Estimated Taxes - click the Start or Update button
Click Start or Update by the taxes you paid