How to get the number of pages when paginating comments?

Try get_comment_pages_count()?

<?php get_comment_pages_count( $comments, $per_page, $threaded); ?>

I’m guessing you’re outside the loop, since you’re calling get_comments(); in that case, you’ll need to pass your $comments object:

$comments =     get_comments(array(
    'post_id' => $post_id,
    'status' => 'approve' 
));
wp_list_comments(array(
        'page' => 1,
        'per_page' => 10,
        'avatar_size'   => 16,
    ), $comments);

$comment_page_count = get_comment_pages_count( $comments );