Install wordpress in github pages

Can we install GitHub using GitHub pages? Or we can add mySQL and database in the wordpress in github?

what do you mean by that ? GitHub Pages is an added service so that a repo can be simply turned to a website,

GitHub Page is browser-side only, just HTML, CSS and JavaScript directly, but you can have a workaround like providing the link on the GitHub Page, but redirecting it to where you can run server-side script

Something you could try (if it fits your needs) is to host (e.g. on your localhost) a Wordpress. then use a plugin that exports the whole blog as static html, and upload it to a Github repository with Pages enabled. However in this case it may be easier to just use something like Jekyll or Hugo.

Theorically you could host a Wordpress and MySQL using Github Actions, but only for testing purposes, without persistent storage and for a limited time.

I say that is we can use a database system in our codes and run the website using Github pages?

GitHub Pages only runs directly browser-side, if there is a technology, language or framework that can communicate ( all browser-side ) to a server that has the access to a database, then your fine,

unfortunately, as far as I know, those full-pledge database are server-side