Heat energy change in adiabatic process We know that when a system gains heat energy from the surroundings through conducting walls of the container , then the temperature of the system also increase till it reaches an equilibrium value .
Whereas in an adiabatic compression of a gas where temperature increases but the change in heat energy of the system is zero .
It must have some value as the state variable of gas i.e. temp. Changed and also we know that                  ∆Q = s.m.∆T
 A: The equation $Q=C_X\Delta T$, where $C_X$ is the heat capacity under some condition $X$, e.g., constant volume, is useful in telling you how much the temperature of a gas rises if you heat it with incoming energy $Q$ under that condition.
However, there are other ways to increase the temperature of a gas; you can compress it, for instance, as you describe in your question. In this case, you did work $W$ on it and therefore increased its energy by $\Delta U=Q+W$ even though $Q$ is zero for an adiabatic enclosure. Its temperature also increased according to the equation of state for an ideal gas $\Delta U=C_V\Delta T$, where $C_V$ is the constant-volume heat capacity (this equation holds for all conditions).
(As I discuss here, the similarity between $Q=C_X\Delta T$ and $\Delta U=C_V\Delta T$ has tripped up untold numbers of thermodynamics students who concluded—incorrectly—that $\Delta U=C_P\Delta T$ for an ideal gas at constant pressure.)
A: The definition of adiabatic expansion is that the heat exchanged by the surroundings and the system is zero.
Also,
∆Q =  ∆U + work done
The work done in an adiabatic process is compensated by the change in internal energy, making sure that no other external factors supply energy to the system.
