Let’s say you have a would-be pullquote like the bland paragraph I have highlighted below. You’d like to format that so it grabs attention (which is the whole purpose of a pullquote), and you’d like to format the starting and ending quotations without having to manually highlight them and apply character styles. GREP styles to the rescue!

1. Create a paragraph style for the body of the pullquote, and a character each for the opening and closing quotes. Make these look however you desire. I created a paragraph style called unironically, Pullquote. My character styles are Pullquote Open and Pullquote Close. 
2. Edit your paragraph style. 
3. On the GREP Style pane of Paragraph Style Options, create a GREP Style. 
4. Set the Apply Style filed to Pullquote Open (or whatever you named yours), and the To Text field to ~{ , which is the GREP code for an opening quotation mark. If you forget the code, you can insert it by clicking on that @ symbol special character menu and choosing Quotation Marks > Double Left Quotation Mark. 
5. Create a second GREP Style configured to show Pullquote Close and using this code: ~} . Your quotation marks should now style independently of the pullquote itself. Tweak your character styles as needed.