How can we use the same search function from phpbb.com ?
Customizing the Google search widget is something you will have to do on your own. However, I agree it is a much better solution than trying to add a new internal search engine.
Actually...it is a better solution than using the internal phpBB search facilities. The internal search system creates the largest tables in the database, and searching is typically the biggest resource requirement (CPU and database) of a phpBB board.