There seems to be something wrong with Moto Connect.
I enabled Moto Connect on my phone and installed the extension on my Chrome Browser. After signing in, I was able to see the SMS, call logs etc... I even tested by sending an SMS, it worked fine by showing the pop-up.
But, after that first notification, I stopped getting notifications for any of my SMS or calls. I even signed out and signed in again, but did not work.
Next day, I started everything fresh. (I removed the chrome browser extension and disabled Moto Connect on my phone. Then, re-did everything). Again, I got the pop-up for the first SMS. After that, notification started appearing.