Setting up Auto Updates in your app with Electron Forge is mostly the same process as described in the Electron docs. Forge enhances your workflow by publishing your app to the right place for you. There are two main ways you can do auto updates.
Open Source Apps - update.electronjs.org
Open Source apps hosted on github.com can use a free auto update service from the Electron team, update.electronjs.org. To use this with Forge, set up the GitHub publisher and add the update-electron-app module to your app.
This setup is going to be around 2 lines of code and a few lines of configuration, by far the easiest way to set up auto updates if you're an open source app.
Host your own update server
If you're not open source or you want slightly more control over your update service you can host your own update server such as nucleus or nuts. See the full list of known Electron update servers in the Electron's Updating Applications docs.
Each update server will have their own configuration for your actual app, but publishing should be done from Forge for most of them: