You can claim the deduction for the Property Taxes you pay in the year you pay them. There is no limitation on how many installments you make towards the Property Taxes per year.
So in your situation described above, you would deduct the two payments you made in October 2017 on your 2017 Tax Return.
The payment you made on December 31, 2016 will need to be added to your 2016 Tax Return. You will do this by filing Form 1040-X to amend the return.
See the TurboTax FAQ below for more information on how to amend your 2016 Tax Return:
https://ttlc.intuit.com/replies/3288565
