As of my last update in September 2021, there is no specific "Google Index API" for pages offered by Google. However, there are various APIs provided by Google that are related to indexing and search functionality.

The benefits of a Google Index API can include:

Google Search Console API: The Google Search Console API allows website owners and webmasters to access data from their Google Search Console account programmatically. It provides valuable insights into how Google crawls and indexes your site, the visibility of your pages in search results, search traffic, and other performance metrics. By using this API, you can better understand how your pages are being indexed by Google and optimize your site accordingly.

Google Custom Search JSON API: This API allows developers to integrate Google Custom Search into their applications. With this API, you can programmatically retrieve search results from Google's index, customize the appearance of search results, and refine searches to target specific content on your site. This can enhance the search experience on your website and help users find relevant content more easily.

Google Knowledge Graph API: The Knowledge Graph API provides access to Google's Knowledge Graph database, which contains information about entities, facts, and relationships between different entities on the web. By leveraging this API, you can enrich your website's content with structured data, making it more informative and easily discoverable by search engines.

Google Indexing API (Deprecated): As of my last update, Google had an Indexing API that allowed webmasters to directly notify Google of new or updated pages for faster indexing. However, this API was deprecated in April 2021, and Google now primarily relies on its web crawling system to discover and index pages.

Remember that Google's APIs and services are subject to change, so it's essential to check the official Google developer documentation for the most up-to-date information on their offerings and guidelines.