The Company's first VAT Return will be for the period 20th April 2020 to 31st July 2020 so what month do I start my Quick Books calculating from?

Hi najcarpentryltd, 

Thanks for joining us on the Community. 

You should set April as the start month, if you set the frequency to quarterly you can then change the end date when submitting by selecting 'Prepare return' > enter an end date.

Alternately if you are submitting through MTD, once you enable the MTD connection the obligation request from HMRC will update on the open return with the correct period start and end dates. 

Thanks