It's quite easy to open a form that will submit to a specific url:
echo Form::model($model, array('route' => array('user', $user->id)))
This will render something like this:
<form action="/user/1">
I want:
<form action="/user/1?q=54">
In other words, how can I change the form method to support the addition of a query parameter, in this case called "q"?
A simpler way of asking this would be: is there a way to generate a URL using Route's so as to include a query paramter?
UrlGeneratorandFormBuildercode and see if they handle appending a query string to a URL.