Should I file my quarterly taxes as a contractor by the 15th? I already filed my 2018 taxes but starting January 2019 I started working as an independent contractor.

If you are referring to your estimated tax payments, then - yes - the first quarter payment is also due on April 15th.  If you are required to make these payments this year due to your employment change, then you will want to make the timely payment or you may have an underpayment penalty on your return next year.

You can print your vouchers or make a direct payment online (make sure to select 1040 ES payments)

https://www.irs.gov/payments

https://www.irs.gov/pub/irs-pdf/f1040es.pdf  (payment vouchers are on pages 9-11)