Change blockquote symbol

The line to the left of the blockquote is actually a border, rather than a character/symbol. I’m afraid that it can’t be switched to use a character instead.

One similar thing you could do to stylize it though, is to switch it to a double border like this:

blockquote {
  border-style: double;
  border-top: none;
  border-right: none;
  border-bottom: none;
}

Here’s how that will look: http://pics.competethemes.com/image/2o220l1Y2f1I

You can add that code to the “Custom CSS” section in the Customizer (Appearance > Customize) to try it out.

The only way to do this would be to manually add the character into the markup like this:

<blockquote>
<span>&raquo;</span>
This is the quote text
</blockquote>

That adds the HTML character code for the double-right arrow symbol: »

Then, you could style it with CSS by using the span element it’s placed in, like this:

blockquote span {
  position: absolute:
  right: 100%;
  font-size: 21px;
}