Before using the API, you need to create one in the Control Panel. Once it is created, you can find the engine's ID in:
Setup > Basics > Details
section of the Control Panel. This is the cx parameter used by the API - developers.google.com 
# Creation deprecated :(
The Custom Search API used to let you Programatically Create Custom Search Engine. It uses standard HTTP requests, such as: GET, POST, and DELETE. All requests must have an Authorization header that contains an authentication token.