WordPress – Query specific category and automatically change category each month

Natively, I would suggest you to create an option page with some options value.

Take a look at this : https://wisdmlabs.com/blog/create-settings-options-page-for-wordpress-plugin/

After created your options, do you logic by getting the options value (= category id) per key (= month number)

Some plugin does that easily (such as ACF, but premium version)