I know we usually use backup as a verb, but if I wanted to use do/make, which one would be correct? I researched on google and found both usages, so I'm not sure which one is right
According to Google Ngram, “make a backup” is by far the most common, but that's in part because it's often used in larger phrases such as “make a backup copy”. “Perform a backup” is less common but also possible. “Do a backup” and “run a backup” are even less common.
This meshes well with my personal experience: I'd normally use “make a backup” in casual speech or writing, or “perform a backup” to be more precise. I wouldn't do or run a backup.