You will need to contact TurboTax support for how to get his TurboTax account User ID and password. They are going to ask for copies of forms which identify both your brother and mother.
Support is open from 5am to 5pm Pacific time, Monday thru Friday.
See this TurboTax support FAQ for contacting support - https://ttlc.intuit.com/turbotax-support/en-us/help-article/account-management/turbotax-phone-number...
Support can also be reached by messaging them on these pages https://www.facebook.com/turbotax/ and https://twitter.com/TeamTurboTax
If a payment was made with the extension request it should be shown in the Tax Account for your mother.
Go to this IRS website for accessing a tax account - https://www.irs.gov/payments/online-account-for-individuals