To activate your desktop notification, you need to Subscribe to the notifications from the bottom right box onYour Dashboard
If you use different computers, make sure you activate it in all as it's based on the device you are on.
First, you need to make sure that the notifications are active. To do so, click on the bell icon, and re-click it to "subscribe to notifications" until you see the "you're subscribed to notifications" message.
Try starting a live chat again.
If it still does not give you a notification, there is probably something that you need to improve in your browser settings. Here are the guides for the different browsers:
If you are using MAC you might also have disabled notifications completely from your browser, here are the guides to fix those issues: