On April 4, in a security notice Canonical published details about a Unity Firefox Extension update for its Ubuntu 12.10 (Quantal Quetzal) operating system.
According to Canonical, this update provides a compatible version of Unity Firefox Extension for Firefox 20.
Mozilla Firefox 20 has just arrived in Ubuntu and the details about this update have been covered in the USN-1786-1 security notice. This particular notification provides the corresponding update for Unity Firefox Extension.
The Unity Firefox Extension update can be applied if you upgrade your system(s) to the latest lxul-ext-unity package, specific to each operating system. To apply the update, run the Update Manager application.
In general, a standard system update will make all the necessary changes. A system restart won't be necessary to implement the changes, but you will need to restart Firefox.