As a Googler I suspect you could, you'd just need permission (possibly per patch).

Googlers contribute to lots of open source projects. Google itself is fairly liberal in terms of what it's willing to share outside. Exceptions tend to be product specific code (you probably couldn't contribute the Hangouts UI and front-end code), particular technologies that are distinct competitive advantages (search ranking, etc.), or things that are Google specific (I work on Google Compute Engine -- there's a fair amount of code that is specific to Google's infrastructure).

And if you really have to ask if such a clause applies to you, you haven't properly read your contract and/or you don't know your rights. People, please educate yourselves, don't let large corporations, even if they claim not to be evil, violate your rights.