Faça isso na linha de comando
uname -i
Se x86_64 estiver listado, você tem sistema de 64 bits. Caso contrário, i386, i586, i686, você tem sistema de 32 bits.
sudo apt-get update
dpkg --get-selections | grep -i qt
Isto lista o seguinte no meu sistema
appmenu-qt:amd64 install
appmenu-qt5 install
libaccounts-qt5-1:amd64 install
libdbusmenu-qt2:amd64 install
libdbusmenu-qt5:amd64 install
libgsettings-qt1:amd64 install
libqt4-dbus:amd64 install
libqt4-dbus:i386 install
libqt4-declarative:amd64 install
libqt4-declarative:i386 install
libqt4-network:amd64 install
libqt4-network:i386 install
libqt4-opengl:i386 install
libqt4-script:amd64 install
libqt4-script:i386 install
libqt4-sql:amd64 install
libqt4-sql:i386 install
libqt4-sql-mysql:i386 install
libqt4-sql-sqlite:amd64 install
libqt4-xml:amd64 install
libqt4-xml:i386 install
libqt4-xmlpatterns:amd64 install
libqt4-xmlpatterns:i386 install
libqt5core5a:amd64 install
libqt5dbus5:amd64 install
libqt5feedback5:amd64 install
libqt5gui5:amd64 install
libqt5multimedia5:amd64 install
libqt5network5:amd64 install
libqt5opengl5:amd64 install
libqt5organizer5:amd64 install
libqt5positioning5:amd64 install
libqt5printsupport5:amd64 install
libqt5qml5:amd64 install
libqt5quick5:amd64 install
libqt5quicktest5:amd64 install
libqt5sql5:amd64 install
libqt5sql5-sqlite:amd64 install
libqt5svg5:amd64 install
libqt5test5:amd64 install
libqt5webkit5:amd64 install
libqt5widgets5:amd64 install
libqt5x11extras5:amd64 install
libqt5xml5:amd64 install
libqtcore4:amd64 install
libqtcore4:i386 install
libqtdbus4:amd64 install
libqtdbus4:i386 install
libqtgui4:amd64 install
libqtgui4:i386 install
libqtwebkit4:i386 install
libsignon-qt5-1:amd64 install
libunity-action-qt1:amd64 install
oxideqt-codecs-extra:amd64 install
qml-module-qt-labs-folderlistmodel:amd64 install
qml-module-qt-labs-settings:amd64 install
qml-module-qtfeedback:amd64 install
qml-module-qtgraphicaleffects:amd64 install
qml-module-qtquick-layouts:amd64 install
qml-module-qtquick-window2:amd64 install
qml-module-qtquick2:amd64 install
qml-module-qttest:amd64 install
qml-module-qtwebkit:amd64 install
qt-at-spi:amd64 install
qt-at-spi:i386 install
qtchooser install
qtcore4-l10n install
qtdeclarative5-accounts-plugin:amd64 install
qtdeclarative5-dev-tools install
qtdeclarative5-qtquick2-plugin:amd64 install
qtdeclarative5-test-plugin:amd64 install
qtdeclarative5-ubuntu-ui-toolkit-plugin install
qtdeclarative5-unity-action-plugin:amd64 install
qttranslations5-l10n install
sni-qt:amd64
A partir desta lista, o pacote que você quer é
sudo apt-get install libqtcore4