On Form 1040-ES, you'll estimate your 2020 income and then pay taxes on a quarter of what you expect to make. The IRS will want you to make quarterly tax payments if you expect to owe $1,000 or more in taxes on your income from a given year.
However, if this is not your first year filing taxes as an independent contractor, then you'll be expected to pay the same amount in taxes as you paid last year.
Additionally, requirements on quarterly taxes can be found here.
Note: The first quarter estimated tax deadline has been moved to July 15th, 2020. The IRS has made this change and others in response to the COVID-19 pandemic. Follow the link here for more information on the IRS date changes.