We study Dolbeault harmonic (1, 1)-forms on compact quotients M = Γ G of 4-dimensional Lie groups G admitting a left invariant almost Hermitian structure (J, ω). In this case, we prove that the space of Dolbeault harmonic (1, 1)-forms on (M, J, ω) has dimension b − + 1 if and only if there exists a left invariant anti self dual (1, 1)-form γ on (G, J) satisfying id c γ = dω. Otherwise, its dimension is b − . In this way, we answer to a question by Zhang., is finite. If the almost complex structure J is integrable, i.e., if (M, J) is a complex manifold, then the numbers h p,q ∂ depend only on the complex structure and not on the metric, being the dimensions of the (p, q)-Dolbeault cohomology spaces by Hodge theory.Whether the numbers h p,q ∂ depend or not on the choice of the almost Hermitian metric ω on a given compact almost Hermitian manifold (M, J, ω) is a question by Kodaira and Spencer, appeared as Problem 20 in Hirzebruch's 1954 problem list [10]. Recently, in [12] (cf. [13]), Holt and Zhang proved that the number h 0,1 ∂ depends on the choice of the metric on a given compact almost complex 4-manifold, answering to Kodaira and Spencer. They also proved that on a compact almost Kähler 4-manifold it holds h 1,1