Can I claim the fourth payment as a deduction on my 2016 federal taxes? No
The state estimated tax payment made in January of 2017 cannot be reported as an itemized deduction on the 2016 Form 1040 Schedule A since it was made after 12/31/2016. It will be entered on the 2017 federal return Schedule A.
The state estimated tax payment made on or before January 17, 2017 can be entered on the 2016 state income tax return as a tax payment.