There is a difference between "paying quarterly taxes" and "filing a tax return".
Quarterly taxes are paid to the IRS each quarter. You also have to pay your state quarterly taxes separately if your state taxes personal income. You only "file" your tax return once a year.
Now the IRS has instructions with worksheets for figuring your quarterly taxes at https://www.irs.gov/pub/irs-pdf/f1040es.pdf. But in my 15 plus years being self-employed I don't waste my time with that. Instead, I just send the IRS 20% of my gross business earnings each quarter. Then come tax filing time it all works out in the wash. For 13 of my 15 years I've always got a refund of some of that quarterly tax payment back. In the two years I actually owed a bit more, it was always less than $500.
If you do the worksheets to figure your quarterly taxes, you'll find that it will come to between 19-21% of the gross business income for the quarter. So I don't waste my time with it and just send the IRS 20% each quarter.
Should she file her quarterly self employment tax in September because I missed the April and June deadlines?
Why wait? The longer you wait, the more the underpayment penalties keep adding up. You should be at least the first two quarters immediately, if not sooner. You can do so "right now" at http://www.irs.gov/payments. Make sure you print the receipt when done and file it with the business records so you'll have it at tax filing time.
From January 2019 to the present, she has earned a total of about $6406.
So I would go ahead and send the IRS 20% of that ASAP, which comes to $1,281. In fact, I'd go ahead and pay $1,400 and consider myself "good" for the third quarter too. If you do that, then when paying on line make "sure" you designate it's a 2019 1040-ES payment for the "THIRD" quarter. Do not bother trying to split things and reporting an amount for each quarter separately. Just pay the total amount due "today" as a third quarter payment. Spliting it up will make absolutely no difference in any late payment penalties if (and only if) any such penalties are assessed. Due to the low amount of business income, there most likely wont' be any penalties assessed. But that depends on the amount of taxes paid through other means, such as your W-2 withholdings each pay period.
