Hello again 🙂
Take a look at one of the default widgets that display simple content, such as the text widget. They are all defined in wp-includes/default-widgets.php. You can use that as a starting point for your widget.
Instead of displaying static content, run a query for quote format posts. Decide how you would select a particular quote. Place the post’s content where the static text would normally go.
The rest of your needs can be dealt with with CSS.