No, websites don't work like that. SSI or CGI scripts assemble a whole page and then send it to the browser which then displays this page. However, you can use JavaScript on your web site. This JS can then make a request to your server when a link is clicked, and replace content in the displayed page by the content from the response.

Such a JavaScript request to the server is called “AJAX”. JavaScript libraries like “jQuery” can help you make the request and insert the response into the page.