If your site is number one in the ranking of Google, Google can show the Sitelinks Search box. Looking something like this:
Setting up Activate Search for the Sitelinks Search Box
To set up your site in order to get this search bar within Google just read this information on Sitelinks Search Box from Google.
If you've taken all the steps here make sure you add a little piece of code to your search bar.
All it has to do is add the 'q' (or whichever value you use) as a 'value' to the search bar.
It should look something like this:
<input type="text" value="<YOUR q-VALUE>" name="Sooqrsearch" id="search" placeholder="Start searching..." />