If all you want to do is confirm the integration is working, you can do this by sending a test notification which is available in the configuration of the webhook. In terms of generating a fake problem, you can possibly create a custom alert and set the threshold to a value that will be exceeded. You can setup your alerting profile to include this custom alert and use this alerting profile for your integration. That way it will generate a problem. You can then delete you custom alert once you verify your integration works as expected.
To me a simple way to emulate a real alert is to configure a synthetic monitor for www.google.com and then verify that the string "ASDFZXCV" is found on the page - since it's not, an alert is generated right away. This is of course also an easy way to test the "resolved" event also, as you can then just edit the monitor and remove that string verification, making the test succeed.