No, the spectrum that is used is well-known and not concealed. You can't conceal that activity is occurring, you can only prevent the SSID from being directly broadcast and encrypt the traffic such that the network can not be spied on without breaking the encryption. The fact that there is radio activity in the spectrum will be apparent to anyone listening in and the encrypted messages will be able to be captured without trouble.
In order for a radio to be able to send a signal, it has to appear above the noise. The only exception to this is when using specialized jamming equipment that allows signals to be sent through it (which is also not WiFi) (generally by not blocking frequencies in a pattern derived from a key), but even then, it is obvious that there is a transmitter there.
You are correct that a good encryption algorithm and key should make the cipher-text look random, but there has to be some kind of way for signal to stand out from random noise. Say that you have a signal that looks random 10011011 and then have actual random noise 01011010. When you combine them, you 11022021, but now since the noise was truly random, how do you know what was noise and what was signal? You can't because you don't have a pattern to the noise.
If you generated the noise, then you could have a pattern and drown out the natural noise, but then we are back to the situation I described for signal jammers, you are going to know they are there because it needs to be enough louder than passive noise to be distinguished from background noise. Such a system would also have to drown out any other signals in the spectrum in the area as they would also be effectively random interference which would make such a device illegal to operate for most purposes in the US due to FCC requirements that a device not cause interference.