After doing a fresh install of Ubuntu 18.04 on my old ASUS laptop, I found that the touchpad right-click does not work. This is because Ubuntu 18.04 by default acts right-click when you perform two-finger click on touchpad.
While my laptop touchpad does not have physical buttons, I have to do two-finger click to perform right-click.
So following steps will show you how to make right-click (clicking on the bottom right area of the touchpad) working again in touchpad without a physical button in Ubuntu 18.04.
1. Open Ubuntu Software, search for and install Gnome Tweaks.
2. Launch Gnome Tweaks and navigate to Keyboard & Mouse settings page.
Under Mouse Click Emulation, simply click enable the second option: “Area – Click the bottom right of the touchpad for right-click and the bottom middle for middle click.”
That’s it. Enjoy!