Short answer, no, push notifications are provided by the browser’s own integrations, so no 3rd party broker is involved.
Technically, there is a server (somewhere, not your Discourse server) that is responsible for delivering messages to your device – our implementation is supported by all browsers that implement VAPID (Sending VAPID identified WebPush Notifications via Mozilla’s Push Service | Mozilla Services). The browser vendor’s VAPID servers are the servers that actually push out the data to your device.