I don't have an answer for the admin bar bubble but here is how you can add it to a NEW admin menu item like this:
(NOTE: Here I'm getting the count for a group of Gravity Form fields, but you can change where you get the count.. displaying it is still the same)
function register_my_custom_menu_page() {
$search_criteria = array(
'status' => 'active', //Active forms
'field_filters' => array( //which fields to search
array(
'key' => 'is_read', 'value' => false, // let's just get the count for entries that we haven't read yet.
)
)
);
// Add the form IDs to the array below, the parent menu will show ALL unread entries for these forms
$notification_count = GFAPI::count_entries( array(1,4,5,6,11,13), $search_criteria );
add_menu_page(
'Full Quote Form submissions', // Page Title
// here we're going to use the var $notification_count to get ALL the form IDS and their counts... just for the parent menu item
$notification_count ? sprintf( 'Quotes <span class="awaiting-mod">%d</span>', $notification_count ) : 'View Quotes',
'manage_options', // Capabilities
'admin.php?page=gf_entries&id=13', // menu slug
'', // callback function
'dashicons-format-aside', // icon URL
6 // position
);
}
add_action( 'admin_menu', 'register_my_custom_menu_page' );