Qt is just a set of C++ libraries. There is no magic there. If Google offers a speech-to-text development kit based on C++ libraries, then you can integrate that just as you would any other library into any other program. If Google's API is a web-based protocol, then Qt has an extensive set of classes supporting most web-based protocols.

What you need to be researching is connecting to the cloud service from a desktop application, not specifically using Qt for it.