you can't do that in WordPress.
However, if you have Google Analytics script installed, you can track all search queries and everything related to them (how many times search was used, which pages, top queries etc)
all you have to do is, select a website profile and edit it
1. Edit Main Website Profile Information
2. Scroll down to "Site Search"
3. click the "Do Track Site Search" radio button
4. Add "s" as query parameter
5. edit any other options as you please
that's all! save the changes and keep track of the data in the website reports > Content > Site Search