$ sudo apt-get update
$ sudo apt-get install libqt4-network 

libqt4-network is installed in your system.

Make ensure the libqt4-network package were installed using the commands given below,

$ sudo dpkg-query -l | grep libqt4-network *

You will get with libqt4-network package name, version, architecture and description in a table

libqt4-network package basic information: