Those conditions are constant volume and constant pressure. This can be explained as given below-

By first law of thermodynamics:

q_v = ∆U + (–w)

q_v = ∆U + p∆V (since, –w = p∆V)

∆V = 0 because volume is constant.

q_v = ∆U + 0

Therefore, q_v = ∆U = change in internal energy

Now, At constant pressure

q_p = ∆U + p∆V

Also, ∆U + p∆V = ∆H

Therefore, q_p = ∆H = change in enthalpy.

Thus, at constant volume and constant pressure heat change is state function because it is equal to change in internal energy and change in enthalpy respectively which are state functions and they are independent of path.