I bought a Q4 AWD back in Oct. 2022, and I experienced the same problem with AC right away. It was not blowing cold air. And warning messages popped up on the console saying that AC system malfunctioned, and also electrical system had problem. I took it back to an Audi dealer. They had the car for around 2 weeks. First they found that refrigerant was low and they thought there was a leak. Eventually, they decided that the AC compressor was faulty. So, they replaced the AC compressor. That fixed the AC problem and it's now blowing cold air.
However, I found another issue with the ventilation control (I posted this issue on another thread). If I turn on turn on the AC ventilation in a certain sequence, the fan control would not work in the manual mode. I took my car back to the dealer and they spent 5 days replacing a connector switch. But, the ventilation control problem remains. I just took the car to the same dealer and they are still working on it.