How to instal a firmware for Monterey system on Macbook air 2017

If the internal SSD is "original Apple" Monterey will install the new firmware on the machine by itself and you can install Monterey internal or on an external drive. If the internal SSD is NOT Apple-original then Monterey will not install internal or external.

Apple distributes firmware via macOS updates. If a firmware update is required it will be installed when installing the update. You no longer need to worry about firmware updates.

However, there are sometimes different versions of macOS for specific hardware models. But this is usually only when Apple releases new hardware after a recent macOS release. There might be a special operating system version just for that new hardware. It's always best to download macOS on the actual model Mac upon which you will be installing it. During the App Store download, specific models might receive a specialized build of the operating system. I don't think that would be the case for your MacBook Air 2017 model.

Steve is correct, if you put a 3rd party SSD into the MacBook Air 2017 you will need to but the original Apple SSD back in so the firmware will update then you can revert back to the 3rd party SSD and continue on. This is a major gotcha that many have run across. Hopefully you still have the original SSD.