I'd go with libcurl any day. It is the most complete library in respect to protocols, ported to numerous platforms, and you really only pay for what you use. Of course you'll also be needing OpenSSL for secure protocols like HTTPS.
On the other hand, if you already have bloated your application with Qt, you might be better served using the QSSLSocket
to retrieve the data and QScriptEngine
to parse the JSON.