You could paste this code directly into the Classic Editor, or into a Block Editor HTML block.
If you don’t want to manually paste the full HTML in every time, you could either create a shortcode or a block that wraps this for you. Something like
<?php
// create shortcode [wpse_quote]
add_shortcode('wpse_quote', 'output_wpse_quote');
function output_wpse_quote($content = null) {
// Wrap the content in divs
return '<div class="quote-wrapper">
<div class="quotes">' . do_shortcode($content) . '</div>
</div>';
}
?>
In the Classic Editor, you would use the shortcode by typing an opening and closing shortcode in brackets, with the quote text between them:
[wpse_quote]Enter the quote text here.[/wpse_quote]
In the Block Editor, you would add a Shortcode Block and then paste that same text in. (Though if you’re using the Block Editor, a block would probably be preferable, because then you can add the actual divs and style them so they look the same on in the Editor as they do on the front end.)