Question: Why the entropy change is not zero in the irreversible adiabatic process?
...while it is defined as the integral of the heat added to the system over its temperature.

Answer: 
If you have an irreversible adiabatic process between two thermodynamic equilibrium end states of a system, there exists no possible reversible adiabatic process between these same two end states. So to get the entropy change for the irreversible adiabatic process, you need to devise an alternative reversible path between the same two end states, and this reversible path will not be adiabatic. On the reversible path, you will have to add heat to the system in order to transition between the same two end states. So the reversible path will give a positive change in entropy.
The statement
$$
dS = \frac{\delta Q}{T}
$$
is only true of reversible processes. The generally true statement is
$$
dS \ge \frac{\delta Q}{T}.
$$
For an adiabatic process, $\delta Q = 0$, but that still allows for $dS > 0$.