How is your "routes.php" file setup? Can you post it? Mind that you are sending a POST request, so with this controller your route should be something like this:

Route::post('contact_request', array('as' => 'contact_request', 'uses' => 'HomeController@test_contact'));

Or in case you registered the controller in the route like this:

Route::controller('home', 'HomeController');

Then your controller method should contain the request verb:

public function postContactRequest()
{  
    // your code
}

In this case, your url would be "contact-request" instead of contact_request, as Laravel already parses it.

Cheers!

This is my routes

Route::post('contact_request','HomeController@getContactUsForm');

Taking your routes.php, this is what you should have in your controller HomeController.

class HomeController extends BaseController {

    /// your other code and stuff...

    public function getContactUsForm() { 
        if(Request::ajax()){
            // return 1;
            echo "yes";
            die();
        }else{
            // return 0;    
            echo "no";
            die();
        }
    }
}

Also, take a look at your javascript. Check your browser console (using the developer tools) for errors.

On a side note, use relative path in an AJAX url, if your route is mywebsite.com/myroute, then your AJAX url should be just "myroute" instead of using the full qualified domain. More info here.