Filter

Change the WP Search URL

add_action( 'template_redirect', 'change_wp_search_url' );

function change_wp_search_url() {
    if ( !empty( $_GET['s'] ) ) {
        wp_redirect( home_url( "/your_page/?your_param=" ) . urlencode( get_query_var( 's' ) ) );
        exit();
    }
}