I don't really know of a way to modify the HTML form to produce that query string from two separate inputs. But in the PHP script that handles the form submission, the URL you're trying to get is basically the equivalent of this:
$_GET['color'] = implode(' ', $_GET);
unset($_GET['car']);
This will change $_GET
to
array (size=1)
'color' => string 'red bmw' (length=7)
as if you had submitted to
http://mywebsite/?color=red+bmw