Let me be more clear. Enthalpy can equal heat when the system does not do any pv work. But the fact that the system does not do any pv work in itself is not a condition for enthalpy equaling heat. Enthalpy change of a system that expands (PV work) can equal the heat transfer between the system and the surroundings.
I don't know what you mean when you say the system is adiabatic. Adiabatic describes processes, not states. Regardless, enthalpy does not necessarily equal work when Q = 0 either.