First, create the stream you'd like to use for Mention Webhook notifications, and subscribe all interested parties to this stream. The integration will use the default stream mention if no stream is supplied in the hook; you still need to create the stream even if you are using this default.

Next, go to your Mention feed and click on your profile in the top right corner, click on Settings in the drop down menu:

From there, select Integrations:

Select the Zapier integration, and click Explore Mention on Zapier!:

Search for "webhooks" in the search bar, and click on Webhooks by Zapier.
Scroll down and look for Add Webhooks posts for new Mentions and click on Use this Zap:

Select your Mention account Id and the alert that should trigger mentions from the respective drop down menus, and click Continue

Select POST on the page that says Webhooks by Zapier Action, and click Continue.

In the URL field, enter a URL constructed like this:

{{ external_api_uri_subdomain }}/v1/external/mention?api_key=abcdefgh&stream=mention

where abcdefgh should be replaced with the API key of your Zulip bot. Select JSON for Payload Type and click Continue:

Congratulations! You're done!
Your messages will look like this: