How Heat is constant in adiabatic process Consider adiabatic cylinder with a piston attached. $Q=0$ in this case and Work done is equal to change in internal energy. If I do work on the piston then its internal energy increases and so its heat because heat is the measure of internal energy of ideal gas. So how we say heat is constant in adiabatic process?
 A: The concept of heat can be quite tricky.  Your cylinder and piston are a system and everything outside the system are the surroundings.  The system has internal energy which is the sum of the kinetic energy and the potential energy of all the molecules which make up the system.  There are two processes by which energy can flow in or out of the system.  One is called working and the other is heating.
If you have an adiabatic change that means that no heat can flow in or out of the system.
Your mistake is to think that because the temperature of the system has increased the amount of heat the system has got has also increased.  That is not true.  The concept that a system has heat stored inside itself is wrong.  The fact that the average kinetic energy of the molecules has increased ie the temperature of the system has increased does not mean that the system has more heat inside it.  In the case of an adiabatic change that increase in internal energy must have been as a result of work being done on the system.
A: Heat is not a state function. We don't say the heat of a system. Instead, it is one way of changing a system's internal energy (which is a state function), the other being work. In adiabatic process, we don't say heat is constant, but heat $= 0$. It is the entropy of the system which is constant in an adiabatic process.
A: Heat is the name given to the flow of energy from one body to another, by virtue of the temperature difference between the two bodies. 
You do not say "this body has so and so amount of heat" You can only say there has been so and so amount of heat flow between these two bodies

If a body is completely insulated perfectly , then there is no possibility of energy coming in or going out (by definition of an insulator). Hence, heat flow is said to be zero. 

In adiabatic systems, the condition is precisely so. Hence if there are systems interacting within an insulated space, (i.e., they cannot derive energy from the rest of the universe) then obviously , they have to work with the amount of energy they have already.

A body has its temperature because of the vibrations of its molecules, if there is flow of heat from it, then the vibrations reduce and its internal energy decreases. This change in internal energy is the only available energy in an adiabatic system to do the required work.
A: In thermodynamics, the term "heat" Q is reserved specifically to refer to thermal energy that flows across the boundary of a system from its surroundings.  If heat flows from the system to its surroundings, then Q is negative.
A: The answer is simple:
$Q=0$ means there is no heat transfer between the system and the surrounding so the only way to supply the energy to the system is by doing work and when you do work you increase the internal energy and hence increase the temperature.  
Optional :
in adiabatic process $PV^\gamma=constant$ where $\gamma=C_p/C_v$ and we know that $PV=nRT$ so we can say that $TV^{\gamma-1}=constant$ so if your ideal gas has been subjected to some work which will decrease it volume. Due to the change in the volume there is the cange in the temperature. And the internal energy is directly related to the change in the temperature have in the system. 
and here system is your ideal gass  
