It sounds like you want to register a web application to Azure Notification Hub to receive notification message. A web application can be as backend to send notification message using SDK in different languages like .NET, Node.js, Java, PHP, and Python. But the answer to receive notification message from web is absolutely not, there is an answer of the exising SO thread Can we register a webapplication to recieve notification from azure notification hub which has answered it.
Azure Notification Hubs are exclusively for push notifications for mobile platforms.
Only one exception is Google Cloud Messaging (GCM) supports Chrome Apps, please see the tutorial Tutorial: Push notifications to Chrome apps with Azure Notification Hubs
For sending and receiving Push Notification in browser, the only way is using Web Push API, you can refer to my answer of this SO thread Azure browser push notification for chrome, firefox ,and safari browser. And there is a Mozilla cookbook site for Web Push to show some examples to help getting started. Then, you can host your web push server on Azure. These Mozilla examples' backend are all using JavaScript based on Node.js, you can get the other web push libraries at this GitHub org web-push-libs if you want to use other languages.