No. Google wants an open internet because search (and hence ads) needs a lot of input. Google gets that input from the open internet (and, notably, shares that input with other search engines). Facebook gets input from 1.8Bn people, and
owns access to it all.
(This is a generalization; Goog certainly owns access to plenty of input: email, docs, and hosting. But it is not, I think, on the scale of FB, because frequent, casual input requires a social context.)
Google's nightmare scenario is that everyone starts to use FB Pages for content, FB ads for that content, and something FB payments to pay for services, etc. In this way FB manages to build a private (but very large) vertically integrated internet. Like Alibaba. There are only three ways to disrupt this trend: the browser, the operating system, and the hardware. Each of these mediates between FB and the user, and each one is Google's opportunity to capture attention. (And this, I think, is why engineering talent is so valuable, just as much to deny FB from making things as to make things yourself.)
The nightmare scenario for the open internet is a Facebook Phone running Facebook OS.