The radio receiver in the YN560 III uses the Yongnuo YN560 communication protocol.
The transmitter in the Neewer trigger does not communicate using Yongnuo's YN560 protocol.
In other words, the transmitter and the receiver do not speak the same language.
Just because they both use the 2.4GHz radio band (like thousands of other short-range wireless radio devices) doesn't mean they speak the same language so that the receiver can understand what the transmitter is saying.
To use the built-in receiver if the YN560 III you need a transmitter that uses the YN560 protocol, such as the Yongnuo YN560-TX or YN560-TX II.
To use the Neewer QC trigger you need to either use a Neewer flash that is compatible with the QC (not all Neewer flashes are, in fact most are not). You might be able to attach a Neewer QC compatible receiver to the hot foot of the Yongnuo YN560 III and trigger it that way.