Forgive me as I'm new to Quickbooks but I'm excited to share a solution that will work for our organization! Maybe it will work for others out there, too!

I will note, this is specifically in Quickbooks Online.

I went to--> Expenses:Vendor:Select Vendor:Edit (edit vendor details) scrolled past address/contact info down to "Additional info", noticed the "Payments" section, updated the terms then ACCOUNT NO. I entered the account number needed on the memo line of the checks. I then tested by going to "New", "Pay Bills", went through that process and saw on the "print preview" page the account number was indeed in the memo line of the check to be printed! It looks like you can add text in that section too, not just numbers. 

For bills that are the same every month, this will be great! For those that change frequently, it's a workaround but definitely NOT a solution.