Online shop opening Sept 30th 2024. Are first quarter taxes due January 15th?

You are correct:  

The 2024 quarterly estimated tax deadlines are:

• 2024 1st Quarter (January 1 – Mar 31): April 15, 2024
• 2024 2nd Quarter (April 1 – May 31): June 17, 2024
• 2024 3rd Quarter (June 1 – August 31): September 16, 2024
• 2024 4th Quarter (September 1 – December 31): January 15, 2025

However; consider if you actually will need to based on your income/profit:   

To determine whether you need to make quarterly estimates, answer these questions:

1. Will you owe less than $1,000 in taxes for the tax year after subtracting your federal income tax withholding from the total amount of tax you expect to owe this year? If so, you're safe—you don't need to make estimated tax payments.
2. Do you expect your federal income tax withholding to amount to at least 90 percent of the total tax that you will owe for this tax year? If so, then you're in the clear, and you don't need to make estimated tax payments.
3. Do you expect that your income tax withholding will be at least 100 percent of the total tax on your previous year's return? Or, if your adjusted gross income (2023 Form 1040, line 11) on your tax return was over $150,000 ($75,000 if you're married and file separately), do you expect that your income tax withholding will be at least 110 percent of the total tax for the previous year? If so, then you're not required to make estimated tax payments.

Quarterly Estimated tax payments deadlines