No, you can't. There's no git running on Google drive.
I would also suggest against Google drive/Dropbox based solutions, and go for a git hosting solution instead. For example Bitbucket which offers some free private repositories. You can find some comparison information about different git hosting sites here.
As people have pointed out (and as OP already knows), you can put the bare repository inside your local Google Drive/Dropbox folder and work with that, however, there are caveats. The cloud services have their own systems for merging conflicts, and that doesn't really work with git. Consider the scenario:
You work with device A offline, push some commits to the bare repository in Google Drive folder, but because you are offline, those changes do not sync to the cloud.
You then forget about it, work with device B online, push commits to Google Drive folder, and those changes do get synced.
Device A becomes online - you now have a conflict in Google Drive.
This is, of course, recoverable, but inconvenient. I therefore recommend to use a solution that is designed precisely for git hosting.
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