Can I have 2 different voltage regulators output 3 different voltages?
Regulators normally do not like to sink current. 

If you connect a load between the 12V regulator and the 9V regulator, then the 9V output will tend to rise out of regulation (to as high as 12V for a heavy load). If a load (to ground) on the 9V regulator can be counted on to always be sufficient, then it may work in some limited situations \$^1\$. 

It's not really an 'output' though, as it's not referred to ground. 




For example, if you put a bypass capacitor between the 12V and 9V outputs then (even with a sufficient steady state load on the 9V output) the 9V output will tend to spike up towards 12V when power is first applied. That could damage something connected to the 9V output.