Is there a possibility to create a copy of the block quote function and then do some basic css additions to it? If yes, then this is what I want to achieve – >
1) A blockquote format but the background is some solid color
2) I am fine to use a shortcode or a direct code to execute it in the post.
if you select the Quote Block ( or any Block ) you can give it a CSS class in Advanced > Additional CSS Class(es). For example red-quote. You can then add some CSS to that class eg.