Modern React relies on hooks, let's rewrite the project using them. See the relevant docs here https://reactjs.org/docs/hooks-intro.html

At the moment jQuery Ajax is used in App.js to grab the file with quotes

componentDidMount() {
    $.ajax({
      headers: {
        Accept: "application/json"
      },
      url:
          "https://gist.githubusercontent.com/camperbot/5a022b72e96c4c9585c32bf6a75f62d9/raw/e3c6895ce42069f0ee7e991229064f167fe8ccdc/quotes.json",
      success: (jsonQuotes) => {
        let quotes = JSON.parse(jsonQuotes).quotes;
        this.props.saveQuotes(quotes);
      }
    });
  }

This can be rewritten using Fetch API.

Before the default case is used in the reducer as an initial state which could be not very tight approach.
It might be better to pass initial state to reducer as default parameter value and return the current state in the default case.

While running the app with npm start, several warnings appears in terminal :

Line 91:19:  Using target="_blank" without rel="noreferrer" is a security risk: see https://html.spec.whatwg.org/multipage/links.html#link-type-noopener  react/jsx-no-target-blank
Line 120:3:  Useless constructor                                                                                                                          no-useless-constructor

At the moment README.md is filled with the default text left from create-react-app. Would be great to replace it with the current project description.

As the repository name was changed to lowercase one needs to update a link in README.md and package.json as well.

After package.json update one also needs to rebuild and redeploy the application.

It is tedious to run build and deploy manually each time. Let's write automation to run these commands for us. GitHub actions can be used for that.

E.g. an existing action https://github.com/JamesIves/github-pages-deploy-action or a custom one.
More about GitHub actions: https://docs.github.com/en/actions