you can use the body class of any page and some css to selectively display (and hide) the submenu.
For example, if your theme supports the page-id body class (as it should...see: http://codex.wordpress.org/Function_Reference/body_class) then you would be able to add css to "display:block" the menu corresponding to a particular page you want it to show upon, and to otherwise hide the nave menu on all other pages with "display:none".
1) Please share your site url so I can help further;
2) Have you coded any submenu's yet that we can use as an example?
3) Do you want some hands-on help to just "get it done"?