If you share a definition page on Facebook from UrbanDictionary.com, the preview image is a screenshot of the definition itself. The same image appears on Google Images. The URL of the image is http://urbandictionary.com/render_definition.php?defid=[the same id as the post]
My question is, how do I make my website do that? I have a website with quotes and would like each quote to be indexed in Google Images as well as display as preview image when shared on social media. I use Wordpress, if that is relevant.
You should be able to use OpenGraph og:image to set an image URL for your page. You need to have a page that will render the image for you.
<meta content="<page_image_url>" property="og:image">
AFAIK, WordPress has a service called mShots that can take snapshots of your page for you. I couldn't find terms of service for it; but if you want to take that route, the following URL will generate a snapshot of http://example.com:
http://s.wordpress.com/mshots/v1/http://example.com/