Here is pseudo code

1. In your OSGi Service class, inject instance of 'InboxNotificationSender'(com.adobe.acs.commons.notifications.InboxNotificationSender)

2. Invoke 'buildInboxNotification()' method on above injected class to get 'InboxNotification' instance
   InboxNotification inboxNotification = inboxNotificationSender.buildInboxNotification()

3. Populate the 'InboxNotification' instance with your details
   inboxNotification.setTitle("Hello World");
   inboxNotification.setContentPath("/my/sample/path");
   inboxNotification.setAssignee("AEM-USER-PRINCIPAL-NAME");
   inboxNotification.setMessage("Greetings of the day!");
   inboxNotification.setNotificationActions("Approve");

4. Invoke 'sendInboxNotification()' method on Step-1 injected object, like

    inboxNotificationSender.sendInboxNotification(resourceResolver,inboxNotification);