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,
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