One way to set Noto Emoji as the default emoji font in elementary OS is to setup some local font config files.
Create the directory and the file itself with:
mkdir -p ~/.config/fontconfig/
touch ~/.config/fontconfig/fonts.conf
And then populate the created file with the following text:
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE fontconfig SYSTEM "fonts.dtd">
<fontconfig>
<match>
<test name="family"><string>sans-serif</string></test>
<edit name="family" mode="prepend" binding="strong">
<string>Noto Emoji</string>
</edit>
</match>
<match>
<test name="family"><string>serif</string></test>
<edit name="family" mode="prepend" binding="strong">
<string>Noto Emoji</string>
</edit>
</match>
<match>
<test name="family"><string>monospace</string></test>
<edit name="family" mode="prepend" binding="strong">
<string>Noto Emoji</string>
</edit>
</match>
<match>
<test name="family"><string>Apple Color Emoji</string></test>
<edit name="family" mode="prepend" binding="strong">
<string>Noto Emoji</string>
</edit>
</match>
</fontconfig>
And finally flush the font cache: fc-cache -f -v
.
If Noto is installed you should now see Noto for emoji unicode blocks in the browser for example, and Symbola filling in where Noto is lacking.