If you use Qt 4.8, which required that you use Qt Webkit, then you're looking at needing to use the QtWebkit Bridge: http://doc.qt.io/qt-4.8/qtwebkit-bridge.html
If you're using Qt 5.7, which also requires that you use Qt WebEngine, then you're looking at needing to use QWebChannel: http://doc.qt.io/qt-5/qwebchannel.html
They allow you to take a QObject class and then share it with the Webkit DOM as a Javascript object. You can then use Javascript and jQuery to call class methods (only ones you expose) of your Qt/C++ QObject class.
BTW, outside of Qt/C++, on the Mac with Objective C, they provide a way to map the AppDelegate to the Webkit DOM using the windowScriptObject, and ObjC lets you use the native Webkit SDK, rather than having to ship another one with your project.
This is great for desktop applications. Unfortunately for iOS and Android, however, both Qt 4.8's Qt Webkit and Qt 5.7's Qt WebEngine are not supported on those mobile platforms. http://stackoverflow.com/a/30662323/105539 I guess on those, you're stuck with using QtQuick/QML interacting with jQuery and C++, I guess? EDIT: On Android and iOS, you can use Qt WebView within a QML application, which uses the native Webkit on each OS. http://doc.qt.io/qt-5/qtwebview-index.html
If I were to develop for iOS and Android, I'd probably use Apache Cordova until something better comes along: https://en.wikipedia.org/wiki/Apache_Cordova#Supported_platf... . One criticism is that Cordova may be slow in some actions, but phones are getting much faster in every release, negating this trouble.