Google can't even tell that the search is coming from DuckDuckGo, as DDG sends this when redirecting to a Google-owned service:

    <meta name='referrer' content='never'>

Interestingly, it doesn't to other networks. For example, Wikipedia, Reddit, and Twitter redirects use this:

    <meta name='referrer' content='origin'>

So they can tell it's from DDG, but not the exact URL (not that it matters because they get the query anyways)