One alternative option would be to use the Cloud Based Lightroom's storage (which doesn't integrate with Google's and assume it a bit more expensive, Google's could indeed be free depending on the content). With the Cloud Based Lightroom your whole Catalog would sync across the Mobile App, Desktop App and Web.
If you want to continue using Lightroom Classics and it's extra features then I assume you cannot Google Photos with it but possibly you can Google Drive.
You can't use any other cloud photo service than Adobe's with any version of Lightroom. The cloud version of Lightroom (what this forum is about) can only use Adobe's cloud storage. If you use Lightroom Classic however, it is possible to use cloud drive storage such as a dropbox folder or google drive to store images. This is a hack though and doesn't always work correctly and can even lead to file corruption so you're better off in general not doing this.