You can display Google Custom Search( CSE ) search form in the menu by adding Google Custom Search( CSE ) search box code in the Google CSE option of plugin displayed in the plugin Menu Search section as shown below.
Please note that you have to only add there Google Custom Search( CSE ) search box code and not search results code and this Google Custom Search will replace any other search form displayed in the menu. Also only using this option will not make it work and you have to manually add Google Custom Search Engine (CSE) script on your site.
Get more information about using Google Custom Search Engine (CSE) in the below pages.
Creating a Custom Search Engine
