If you are connecting a public bucket, simply fill out the bucket name in the integration modal. After that your bucket will be mounted into your filesystem.
If you need help with making a bucket public, here is a tutorial
In Google Cloud Platform, create a service account and download the key (guide) or re-use an existing one.
In Deepnote, create a new Google Cloud Storage integration and enter the name of the bucket. Then copy the contents of the service account into the empty field.
After the integration is created, you can connect it to any of your projects. The bucket will be mounted to your file system where you can easily access it.