I contribute to my ROTH IRA on a monthly basis. Do I count the first four months of 2017 as my 2016 contribution since we have the ability to contribute up until April 18?

It depends.

You would only count your 2017 payments (up to April 18th, 2017) as part of your 2016 contributions if you actually designated that these contributions made in 2017 will be part of your 2016 contributions. If you did not specifically designate that these 2017 contributions be considered part of your 2016 contributions, then you will not include these 2017 contributions as part of your total 2016 contributions.

Please refer to IRS - Roth IRAs - Publication 590 for more information about Roth IRAs.