Are you using a child theme? If so, look in your child theme’s style.css file and find the blockquote styles and change the code to something like below, or add the custom CSS into any additional CSS areas that may be included in your theme’s Customizer options to override the stylesheet.
To take inspiration from the referenced example, you can do something like…
blockquote {
background-color: black;
color: white;
border-radius: 4px;
font-size: 2rem;
font-style: italic;
}
Note that the code above doesn’t include the quote character in the background. There’s a couple of approaches to accomplish that—either using a background-image + background-position, or using a ::before pseudo-element (depending on what icon fonts you’re using).