One single instance of QNetworkAccessManager is enough for an entire application. For example, you can create an instance of QNetworkAccessManager in your main window class and pass a pointer to it to all the other places where it's needed. For ease of use, you can also create a singleton and access the manager through that.
A simple template-based approach to create a singleton will look like this (as a header file):
template <class T> class Singleton { public: static T& instance() { static T static_instance; return ...