If you sign up an verify the site with Google Webmaster Tools you can submit a sitemap. It's a simple XML format and pretty easy to generate each time your site is updated, this allows you to quickly submit large numbers of URLs to Google for indexing (though bear in mind that adding a URL to a sitemap does not mean Google will index it, rank it or speed up it's submission - there are no guarantees!).
<?xml version="1.0" encoding="UTF-8"?>
This is about as close as they come to such an API, my guess is your client has heard of this and is giving you the Chinese whispers version of the explanation they received.
That said blogging software like Wordpress does 'ping' Google Blog Search and other sites with new posts - see http://blogsearch.google.com/ping / http://blogsearch.google.com / 2012 Ping List, this service does appear to list blog postings faster than the standard Googlebot.
Blog Search did have an API for displaying results on websites but this was depreciated in 2011 - your client could also have heard about this as well and perhaps is conflating the faster listing of blog postings with the listing API thus the mangled explanation you've received.