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.

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.