As said, Joomla is based on PHP and generally uses MySQL as database. Since you've mentioned Mono, I assume you want to install this on a Linux system, in which case Joomla would be a good choice.
However, you also want to use ASP.NET, which would work much better on a Windows system, not Linux with Mono. Even though ASP.NET development is possible with Mono, you will still have to deal with lots of .NET functionality that hasn't been (completely) implemented for Mono. I would try to install VMWare (or other virtual machine software) on this Linux NAS device with a Windows Virtual machine running inside it. But I doubt this QNAS device would have enough memory and processing speed for this. And installing Windows on this device isn't a good idea either, in my opinion. So you will have a few bumps ahead when you want to develop some ASP.NET code for this device.
However, combining ASP.NET and PHP together on one system should not be a problem. And I wouldn't really worry about the Joomla and other PHP stuff on this device. I'd be worried more about the .NET development.
But consider this option: develop and run your ASP.NET applications on your Windows system, turning this system in a second server. Then get these two to work together, where the QNAS device would be communication with ASP.NET (WCF) web services to retrieve data, with the QNAS device itself using PHP, not ASP.NET as development language.
Combining two web development languages is a bit complex, but it will also have a very good educational value. Basically, once you understand how something works in one language, it also becomes easier to understand in the other language.