My Brother printer not working in macOS Big Sur(11.3.1)

Brother recommends using AirPrint with Big Sur. Unfortunately, your model does not support AirPrint. As an alternative, Brother recommends you download Brother iPrint&Scan from the Apple App Store. Note: Most reviews for the Brother iPrint&Scan software are not very good.

When viewing the Downloads webpage, there is no printer driver available for macOS 11. Therefore I would assume Brother will not be releasing a printer driver for your current macOS 11.3.1.

The good news is that the Brother MFC-1910W fully supports macOS 11 "Big Sur", as you can see in the printer's Supported OS page:

The bad news is that the printer doesn't support AirPrint, a feature in macOS (and iOS) that supports printing without installing printer-specific drivers. In my experience, AirPrint is especially convenient for MFPs because of its support for scanning.

I'd recommend that you return the printer and purchase one that supports AirPrint. Check Apple's AirPrint device list for a complete list of supported devices.

If that's not an option, you will have to install iPrint&Scan from the Mac App Store (the Mobile Print/Scan Guide for Brother iPrint&Scan (Apple Devices), although it applies to iPhones/iPads, may be helpful).