Yes, you can develop software on Ubuntu, that itself will run on Ubuntu, in C#. Both Mono and .NET Core support GNU/Linux systems like Ubuntu. (You can use them on other distros, too, like Debian, Raspbian, Fedora, CentOS, Arch, Gentoo, and so forth.) The Unity3D game engine also supports Ubuntu, as you probably know. Unity3D embeds Mono.
A number of packages for Ubuntu, installable with Ubuntu's package manager from officially community-supported software sources, are written in C# and use Mono. This includes the music player Banshee, the notetaking app Tomboy, the raster graphics editor Pinta, and the password manager KeePass (since version 2).
Most text editors, such as Gedit, Vim, and Emacs, have syntax highlighting for C#. MonoDevelop and Visual Studio Code are two popular integrated development environments that run on Ubuntu and support C# development.