zlib is contained in the core Qt libraries. If you want to use the zlib functions in a Qt program, you only have to include zlib.h which is in src/3rdparty/zlib. See e.g. the implementation of QByteArray in src/corelib/tools.
If you want to use quazip, just add the library to your project. It is based on the Qt libraries. Take care to build the correct qyazip library that corresponds to your Qt installation.
You get the correct include path by adding the following line to your project file:
INCLUDEPATH += $$[QT_INSTALL_PREFIX]/src/3rdparty/zlib
For Qt5, see Thorbj?rn's comment: it is sufficient to use #include <QtZlib/zlib.h>
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