I've been using Tapatalk HD for iPad for quite some time and I've never received a notification from it. I use Tapatalk 2 on iPhone and that always gave me notifications.
I looked in the App store at '2' and 'HD' and it appears to me that '2' is newer than 'HD' and the last update to 'HD' was in December 2012, while the newest update to '2' was May of 2013.
I deleted 'HD' and installed '2' on the iPad and logged in. I've already received a notification on iPad with '2' installed. So I guess 'HD' is not the way to go on iPad if one wishes to receive notifications.
The only push notifications I've been getting over the last week has been new threads created in a forum and PM notifications. No quote notifications have been coming in at all. This is the same for both iPhone and iPad.