MQTT3.1.1 Implementation in QT
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No, Qt framework does not have MQTT implementation.
Maybe some third party libraries like "this":https://github.com/slimpp/qmqtt one -
This library is 2 years old. I don't know when MQTT 3.1.1 was accepted so it is possible that the library is a bit outdated.
However if this library implements the required protocol and satisfies your requirements then you can link it as static or shared library to your project.