In the Apple Notes app, I see an ICLOUD folder and a GMAIL folder. When I select "Edit" -> "Move All" the only option is to select ICLOUD. GMAIL isn't an option.
I am able to copy notes to the ICLOUD folder (but I have a bunch of notes and that seems annoying) and I am also able to "Save to Files" my Drive account. In both cases, the note does reach my google account so I don't think it's an account issue.