Is is possible to change the emoji set on OS X?

Not Google's emoji, but similar: if you have an font file (in .ttf format) containing emoji, you can replace Apple's emoji font with that font, as described in the GitHub repo for Emoji One:

Mac OS Instructions: Using the latest OS (El Capitan), I was able to load this emoji file in less than a minute. The original emoji ttf is located in system/library/fonts, do not touch this. You can safely upload the renamed file (from emojione-apple.ttf, to Apple Color Emoji.ttf) to the /library/fonts folder. That file will override the default. This was my experience, but yours may vary.

I tried this method with Google's Noto fonts, but unfortunately, OS X (El Capitan) wouldn't install the font due to some sort of incompatibility. In the meantime, though, I've successfully gotten Emoji One to work following the instructions, which may work if Noto Color Emoji ever works on OS X.

Edit: As pointed out by Evan Purkhiser in the comments, all you need to do is download the Emoji One Apple font file, rename it to Apple Color Emoji.ttf, and save it in ~/Library/Fonts to override the system emoji. I just tried this on Sierra and it worked immediately.