QSO is an amateur (ham) radio (or "Q") code used when one operator is making initial contact with another operator. The receiving operator typically responds with a greeting and a signal strength report before a longer conversation begins. Q-codes, like the better known ten-codes (10-4) used by citizen's band radio operators, are used to convey information using brief phrases. Once used widely by maritime, law enforcement, rescue, and aviation personnel, their use is now largely confined to military and amateur radio operators.
Fermi's paradox is the apparent contradiction between the high probability that non-human intelligence must exist somewhere in the vastness of the universe and the lack of evidence for its existence.
The code that Root sends to Shaw means "four-alarm fire", which is part of the quote of Shaw: "But, you and me together would be like a four alarm fire at an oil refinery."
QSO also stands for "quasi-stellar object" or "quasar" - extra-terrestrial radio sources that were originally thought to be a sign of advanced, intelligent civilizations.
QSO is part of the 'Q code' system, which was invented for early radio-telegraph communication. The Q code system was created to make early radio communication more efficient. Back in the day only Morse code was used, which was quite inefficient. The Q codes were 3 character codes that related to commonly asked questions and or replies. The code 'QSO' means: "Can you communicate with ... directly (or through...)?"