Is there a Qt install path variable that I can use in the .pro file?

Another solution (may be not as fancy as above with $[QT_INSTALL_LIBS] but I used it already for quite a long time:

TEMPNAME = $${QMAKE_QMAKE}
QTPATH = $$dirname(TEMPNAME) 

then you can reference it like this (for example to access some private headers) or to copy things:

INCLUDEPATH += $$QTPATH/../../Src/qtbase/src/sql/kernel

For Qt4 and Qt5, looks like $$[QT_INSTALL_LIBS] is what you want? Can't confirm first-hand this works though.

See https://forum.qt.io/topic/65778/qmake-and-qt-installation-root-directory/2 and http://doc.qt.io/qt-4.8/qmake-advanced-usage.html.