You'll not pay anything on a quarterly basis if you have 0 income. However, you'll still need to submit your Schedule C form at the end of the year. Let me provide you with some information on how estimated quarterly taxes work.
QuickBooks Self-Employed calculates the estimated taxes based on these three factors:
Your self-employment income and deductions
Your tax profile or tax setup
To learn more about the calculations of your estimated taxes, here are great resources that you can check: