Get page by its ID and display it on another one

I am not getting anything, an empty page. Am I missing something?

Yes, you are.

$query->the_post() does not display anything — it just moves the pointer/key in the posts array ($query->posts) to the next one, and setups the global post data ($GLOBALS['post']). So yes, in response to your answer, the_content() would be what you would use to display the content of the current post in the loop.

But actually, you could simply use the page_id parameter like so, to query a post of the page type and having a specific ID:

$args = array(
    // Instead of using post_type and post__in, you could simply do:
    'page_id' => 1149,
);

Also, you should call wp_reset_postdata() after your loop ends so that template tags (e.g. the_content() and the_title()) would use the main query’s current post again.

while ($query->have_posts()) {
    $query->the_post();
    the_content();
}

wp_reset_postdata();

And just so that you know, you could also use get_post() and setup_postdata() like so, without having to use the new WP_Query() way:

//global $post; // uncomment if necessary
$post = get_post( 1149 );
setup_postdata( $post );
the_title();
the_content();
// ... etc.

wp_reset_postdata();