Multiple Block Quotes without using HTML

you can create shortcode for that

1.make a shortcode unsing add_shortcode(functions.php)

add_shortcode('bquote','ravs_blockquotes_func')    
    function ravs_blockquotes_func( $atts, $content="" ) {
             extract( shortcode_atts( array(
                  'style' => 'style1'
             ), $atts ) );
        return '<blockquote class="'.$style.'">'.$content.'</blockquote>' 
        }

2.now just simply define quote in shortcode(post or page editer)

[bquote style="style2"]your quote[/bquote]

it’s output in html like

<blockquote class="style2">your quote</blockquote>

3.create as many as class for blockquote style you want.it’s easy to use and understand.

Important Link:

WordPress Shortcode API

WordPress shortcode complete guide