Bash
sudo apt-get install QtSerialPort
E: Unable to found QtSerialPort
 

Но если сам Qt ставится не из реп, а из инсталлятора - то там должен быть QtSerialPort (хотя не уверен, нужно проверять тебе самому ).

Bash
xokc@astra-client:/usr/lib/x86_64-linux-gnu$ ls libQt5*.so.5.3.0
libQt5CLucene.so.5.3.0             libQt5Multimedia.so.5.3.0         libQt5Svg.so.5.3.0
libQt5Concurrent.so.5.3.0          libQt5MultimediaWidgets.so.5.3.0  libQt5Test.so.5.3.0
libQt5Core.so.5.3.0                libQt5Network.so.5.3.0            libQt5WebKit.so.5.3.0
libQt5DBus.so.5.3.0                libQt5OpenGL.so.5.3.0             libQt5WebKitWidgets.so.5.3.0
libQt5Declarative.so.5.3.0         libQt5PrintSupport.so.5.3.0       libQt5Widgets.so.5.3.0
libQt5DesignerComponents.so.5.3.0  libQt5Qml.so.5.3.0                libQt5X11Extras.so.5.3.0
libQt5Designer.so.5.3.0            libQt5Quick.so.5.3.0              libQt5Xml.so.5.3.0
libQt5Gui.so.5.3.0                 libQt5Script.so.5.3.0
libQt5Help.so.5.3.0                libQt5Sql.so.5.3.0
 

Bash
root@astra-client:/usr/include/x86_64-linux-gnu/qt5# ls
QtConcurrent  QtDBus  QtNetwork  QtOpenGLExtensions  QtPrintSupport  QtSql   QtWidgets
QtCore        QtGui   QtOpenGL   QtPlatformSupport   QtSolutions     QtTest  QtXml
 

Очевидно, что что-то делаешь не так.. 

Bash
g++ -c -g -O2 -fstack-protector --param=ssp-buffer-size=4 -Wformat -D_FORTIFY_SOURCE=2 -O2 -fvisibility=hidden -fvisibility-inlines-hidden -std=c++0x -fno-exceptions -Wall -W -D_REENTRANT -fPIC -DQT_NO_MTDEV -DLINK_LIBUDEV -DQT_BUILD_SERIALPORT_LIB -DQT_BUILDING_QT -DQT_NO_CAST_TO_ASCII -DQT_ASCII_CAST_WARNINGS -DQT_MOC_COMPAT -DQT_USE_QSTRINGBUILDER -DQT_DEPRECATED_WARNINGS -DQT_DISABLE_DEPRECATED_BEFORE=0x050000 -DQT_NO_EXCEPTIONS -D_LARGEFILE64_SOURCE -D_LARGEFILE_SOURCE -DQT_NO_DEBUG -DQT_CORE_LIB -I/usr/lib/x86_64-linux-gnu/qt5/mkspecs/linux-g++-64 -I/home/xokc/test/qt-qtserialport/src/serialport -I/home/xokc/test/qt-qtserialport/src/serialport -I/home/xokc/test/qt-qtserialport/include -I/home/xokc/test/qt-qtserialport/include/QtSerialPort -I../../include -I../../include/QtSerialPort -I/home/xokc/test/qt-qtserialport/include/QtSerialPort/5.3.0 -I/home/xokc/test/qt-qtserialport/include/QtSerialPort/5.3.0/QtSerialPort -isystem /usr/include/x86_64-linux-gnu/qt5/QtCore/5.3.0 -isystem /usr/include/x86_64-linux-gnu/qt5/QtCore/5.3.0/QtCore -isystem /usr/include/x86_64-linux-gnu/qt5 -isystem /usr/include/x86_64-linux-gnu/qt5/QtCore -I.moc -I. -o .obj/qserialport.o /home/xokc/test/qt-qtserialport/src/serialport/qserialport.cpp
In file included from /home/xokc/test/qt-qtserialport/src/serialport/qserialport.cpp:45:0:
/home/xokc/test/qt-qtserialport/src/serialport/qserialport.h:48:44: fatal error: QtSerialPort/qserialportglobal.h: Can't found file or directory
compilation terminated.
make[2]: *** [.obj/qserialport.o] Error 1
make[2]: Leaving directory `/home/xokc/test/build/src/serialport'
make[1]: *** [sub-serialport-make_first] Error 2
make[1]: Leaving directory `/home/xokc/test/build/src'
make: *** [sub-src-make_first] Error 2

DOS
Package: libqt5serialport5
Source: qtserialport-opensource-src
Version: 5.3.0-1
Architecture: amd64
Maintainer: Debian Qt/KDE Maintainers <debian-qt-kde@lists.debian.org>
Installed-Size: 138
Pre-Depends: multiarch-support
Depends: libc6 (>= 2.2.5), libqt5core5a (>= 5.2.0), libstdc++6 (>= 4.1.1), libudev0 (>= 0.140)
Multi-Arch: same
Homepage: http://qt-project.org/
Priority: optional
Section: libs
Filename: pool/main/q/qtserialport-opensource-src/libqt5serialport5_5.3.0-1_amd64.deb
Size: 32718
SHA256: a0c789655f09ce93449ea1e4a2fafc7756642d7acd239697f8747dd9cea22784
SHA1: 131fb70a369f24b706e6cc5f18c923eeb5570fe4
MD5sum: 35c7ea305a6ea95cecac24f87113434c
Description: Qt 5 serial port support
 Qt is a cross-platform C++ application framework. Qt's primary feature
 is its rich set of widgets that provide standard GUI functionality.
 .
 This package includes the library to use serial ports from within Qt 5.
Description-md5: 495baaf3e432acf5b4e4136a5d0e645a

Bash
sudo apt-get install libqt5serialport5