Can I use Google Emoji on mac?

Yes and no. You can change the default emoji font used by Mac OS X, but I don't think you can change it to Google's Noto fonts, although that may depend on your version of OS X.

Basically if you have a font file in .ttf format that contains emoji, you can replace Apple's emoji font with that font using the following steps:

1. Ignore the default location for Apple's emoji located in system/library/fonts. In other words, leave this alone.
2. Obtain another emoji font file. HINT: The emojione font is similar to Google's.
3. Rename your other emoji ttf font file to Apple Color Emoji.ttf
4. Move the newly renamed font file to the /library/fonts folder.

I know the above process works for El Capitan, but your situation may be different.

Yes, you can!

--EDIT-- The below still works, BUT, turns out, JoyPixels (open source emoji and the closest Google emoji / Noto Font replacement) has multiple versions of their emoji set including a .ttc file directly for download on their website for free that you can simply download and install as a font easily: https://www.joypixels.com/fonts

Fonts are in a .ttf file format, but Apple compiles those files using a SBIX format and others use other methods, typically CBDT/CBLC is used. Because of this, even though you have a .ttf file, it has to be compiled in the SBIX format. There is a user (vXBaKeRXv) on Github who has compiled multiple emoji fonts using the SBIX format. There's a Github thread and a Medium article on what to do. Quoting from the Medium article:

The Step-by-Step

1. Head over to this Github repo and download the .deb archive that contains your emoji set of choice.
2. Extract/open the .deb archive. Once inside, extract the archive named data.tar.gz. You’ll now see a folder called var. Inside that is a folder called mobile. Keep on going deeper and you’ll eventually reach your destination: a font file named [email protected].
3. Rename [email protected] by putting spaces between each word and removing the ending, so that the new file name is Apple Color Emoji.ttf.
4. Keeping that Finder window open, start up Font Book (the built-in app where you can manage all the fonts installed on your device). In the left side panel at the top, select the ‘Computer’ option to show all the fonts you’ve installed as well as the default system fonts. If you scroll down the list of fonts, you should see an entry called Apple Color Emoji close to the bottom. This is the font file you’ll be overwriting, but don’t worry, this process is easily reversible!
5. Drag the Apple Color Emoji.ttf from the Finder window into Font Book. You’ll be prompted to install the font. It may warn you about it being a duplicate, but we already know that, so keep on going. Once you finish installing the new emoji font, you’re good to go! Try out your new emoji set by opening up your browser of choice, selecting a text field, and press CTRL+CMD+SPACEBAR to quickly bring up the emoji picker.

For step 2, this works on a Mac to unpack a .deb file without installing it: ar -x path/to/deb/file.deb

No, you can't.

Apple uses different type of ColorEmoji font, SBIX if I'm not mistaken, Google uses another different CBDT/CBLC format, so they are incompatible for each other.

If you want to use Google's NotoColorEmoji, you need to decompile ttf file, and recompile it as Apple compatible version.