Is there an API available for Google's Related Search Queries?

No, Google doesn't have any public API that you can use to retrieve a list of related URLs programmatically. But if you can settle for an alternative, then Bing has a public API for a similar purpose:

http://api.bing.com/osjson.aspx?query=clooney

Unfortunately, Google has sunset the Related Queries API. There are similar and semi-similar APIs though.

I am not affiliated with any of these, and haven't used any of them.

Similarweb provides an API to find domains that are similar to passed domains. It's not the same as search terms, but if you know which domains you wish to compare, that is an option. It is not a free service though.

Another pay service is from KeywordDiscovery and their related keywords.

Yahoo! provides a related search service. It is not free, and will return 10 results per query. This is part of their BOSS search product.

The Freebase API is an open source knowledge graph that contains the publicly available part of the knowledge base that Google uses to populate there related queries list.

It is actually released by Google however contains only a subset of there full knowledge graph due to legal restrictions from some of there data sources.

Search results can be queried programmatically using HTTP and JSON or there is a variety of programming language client libraries

Here is the default JSON result for the 'clooney' example given in the question. Which could be filtered via the API and used as seeds for branching the search depending on the usecase.