The Gmail Message Tray GNOME Extension does exactly what you want: shows you notifications of emails in the GNOME message tray (aka ‘notification centre’ aka ‘date/time indicator’).
One of the main reasons I use a desktop email client such as Thunderbird or Geary is the ability to receive notifications on the desktop as and when new email arrives.
But if you use Gmail (and chances are you do) and prefer using the web client in your browser, configuring an entire desktop email client just to stay informed can seem overkill or unnecessary.
Shows Gmail Notifications in GNOME Shell
If all you want to see are desktop notifications from Gmail on the GNOME desktop, you can, using the Gmail Message Tray GNOME extension.
You don’t even need to idly enter your Gmail password or account information to get set up, either as the extension uses Gnome Online Accounts to login to Gmail and check your incoming email.
When you have unread mail you’ll see a desktop toast telling you how many messages are waiting, plus individual notifications for each mail in the Message Tray. Click on a notification to get swooshed over to Gmail in your default web browser.
The downside to the extension is that it’s not (currently) very smart, and it doesn’t work like push notifications.
You can configure how often the extension checks for new mails but, every time it checks, it will fetch and display notifications for every unread message, regardless of whether you’ve already seen a notification for it.
On the plus side when you have no new emails the extension will tell you exactly that, which may seem like an annoying feature but it reassures you that the extension is actually working.
As notification extensions go this one is perhaps better suited to those of you who keep on top of your emails. For users with large inboxes or who treat their unread messages as their to-do list (like I do) this extension can be too overwhelming.
Gmail Message Tray works on GNOME 3.24 only, and may require
gir1.2-gconf to be installed on your system.