Thank you. As indicated by Old Toad, a Brother MFC-J475DW is mentioned in About AirPrint - Apple Support as a model supporting the driverless AirPrint technology. This means that it should be possible to use it without drivers if it is connected to the same network (Wi-Fi router) as the computer. An AirPrint-enabled printer is compatible with macOS (Big Sur).
However, for some reason, nothing about Big Sur here: https://support.brother.com/g/b/oslist.aspx?c=us_ot&lang=en&prod=mfcj475dw_us_as
You could try to reset the printing system and re-add the printer (if possible, using AirPrint):
Reset the Mac printing system to solve a problem - Apple Support
Add a printer on Mac - Apple Support
If the printer is connected via USB, drivers will be needed. We have seen a number of cases where a macOS Catalina 10.15 driver could be used with Brother printers even under macOS Big Sur. You may want to test the 1.6.0 printer driver here: https://support.brother.com/g/b/downloadlist.aspx?c=us_ot&lang=en&prod=mfcj475dw_us_as&os=10060
As with any test, things do not always work. Software conflicts cannot be ruled out. Data may be lost. So, before carrying out tests, make sure that you have multiple copies of all important files (documents, photos, et cetera).