iMac (2013) OS upgrade for Brother printer installation

An iOS version of what exactly? An app? Are you able to print from an iPhone? How do you do that?

Providing that Brother printer's exact model designation would be most helpful in answering your question.

John Galt wrote:

What specific model Brother printer do you have? You need to provide its exact model designation, information you might be able to find printed on its back label or elsewhere on the device.

Most Brother printers are AirPrint-compatible. If yours is, then you don't need an app, and no particular macOS version is a prerequisite.

Good news, that printer is Apple AirPrint-compatible: About AirPrint - Apple Support, so you don't need any app for the iMac. Just add it to Printers & Scanners as an AirPrint printer according to Connect an AirPrint printer to a Wi-Fi network - Apple Support. You have to explicitly choose AirPrint.

Printing from an iPhone is described here: Use AirPrint to print from your iPhone or iPad - Apple Support. Of course you can still use the Brother iPrint&Scan iOS app if you so choose, but it's not necessary. Everything it does can be accomplished on a Mac without any special software.

What solution are you seeking? If Brother says a particular operating system is required, then perhaps it is.

On the other hand perhaps you don't need their installation program. What specific model Brother printer do you have? You need to provide its exact model designation, information you might be able to find printed on its back label or elsewhere on the device.