Ubuntu (18.04) uses IBUS to be the IMF (Input method framework).
And it manages the IMEs (Input Method Editor), like MOZC.
Install:
- ibus:
sudo apt install ibus.
- mozc:
sudo apt install ibus-mozc mozc-utils-gui.
Navigation
To see the configuration window:
- Click on
Gnome Status menu (Gnome's top-left bar) > IBus menu (i.e. keyboard/input language selector) > Select Japanese Mozc input method.
- Again, click on
Gnome Status Menu > IBuz > tools > Properties > Mozc Settings window.
Screenshots

Alternatives
You can install/use another IMFs and japanese IMEs.
IMEs
Others IMEs to substitute MOZC when typing japanese are:
- Anthy:
sudo apt install ibus-anthy.
- KKC:
sudo apt install ibus-kkc.
- SKK:
sudo apt install ibus-skk.
IMFs
One suggestion to substitute IBUS is FCITX:
- Fcitx:
sudo apt install fcitx.
And you will need to install all IMEs again, like:
sudo apt install fcitx-mozc.
sudo apt install fictx-kkc.
sudo apt install fictx-skk.
A list of IMF can be found in List of input methods for Unix platforms and Linux input method framework brief summary.