In short: I think the answer here is "no", you can't use the web fonts from Google as display fonts in Emacs.
The feature "download and use arbitrary fonts over the network" is deemed useful enough to implement for apps whose purpose is "download and display arbitrary text and multimedia content over the network", notably browsers, but not for almost all other applications, including Emacs, which rely on locally installed fonts.
woff2 and Microsoft's
These file formats are compressed representations of fonts, so that they can be sent with web pages that use them. The font files are interpreted within the web browser and used primarily to display letters in a page on a screen.
Emacs runs in either a desktop (like Windows, Gnome OS X) or as a terminal application. The fonts it can use are those that are provided by that desktop or terminal. These are the ones that are installed locally in whatever way that system provides. In modern desktops,
ttf are the commonest file formats.
Basically, Emacs is here the same as any other desktop (or terminal) application in using the fonts offered by that environment. It's browsers that are "special" in knowing how to use web fonts.
Many fonts that are offered "free" as web fonts are also available for download and installation as
otf. Using Emacs's menus (e.g. "Options > Choose Default Font") may make testing out different display fonts easier.