This article will not go into the details of how to create the Custom Search Engine (CSE). Google has a good tutorial on that already. Check it out here. The prerequisite for this post is that you have created and configured the CSE as a results only page.
After the template page is created, the index.php file from FlatPress needs to be modified. Below are the modified lines from the file, with the original include call and return statement commented out. The return statement is changed to set the module variable to the name of the template file. This has the effect of rendering the search results page whenever the 'q' URL parameter is specified. Since CSE uses 'q' as the default parameter, this works right out of the box.