Here is an example to think about: Let $S^6 = \mathrm{G}_2/\mathrm{SU}(3)$ be the $6$-sphere endowed with its $\mathrm{G}_2$-invariant almost Hermitian structure. There is a $\mathrm{G}_2$-invariant (special) Hermitian connection $A$ on the tangent bundle of $S^6$ whose curvature is of type $(1,1)$, but the only $\mathrm{G}_2$-invariant almost-complex structures on $S^6$ (and there are exactlty two) are not integrable.

The point is that the (special) Hermitian connection $A$ has torsion of type $(0,2)$, and that is what prevents the almost-complex structure from being integrable.

Of course, there might still exist some complex structure on $S^6$ (whether this is so is not currently known), but the existence of an Hermitian connection on its tangent bundle with curvature of type $(1,1)$ has proved of no help, so far, in finding a complex structure on $S^6$.

Here is an example to think about: Let $S^6 = \mathrm{G}_2/\mathrm{SU}(3)$ be the $6$-sphere endowed with its $\mathrm{G}_2$-invariant almost Hermitian structure. There is a $\mathrm{G}_2$-invariant (special) Hermitian connection $A$ on the tangent bundle of $S^6$ whose curvature is of type $(1,1)$, but the only $\mathrm{G}_2$-invariant almost-complex structures on $S^6$ (and there are exactlty two) are not integrable.

The point is that the (special) Hermitian connection $A$ has torsion of type $(0,2)$, and that is what prevents the almost-complex structure from being integrable.

Of course, there might still exist some complex structure on $S^6$ (whether this is so is not currently known), but the existence of an Hermitian connection on its tangent bundle with curvature of type $(1,1)$ has proved of no help, so far, in finding a complex structure on $S^6$.

Here is an actual counterexample: S. T. Yau has shown that there are compact parallelizable 4-manifolds $M^4$ that support no complex structure. See S. T. Yau, Parallelizable manifolds without complex structure, Topology 15 (1976), 51–53. Since the tangent bundle is trivial, there is an almost-complex structure $J$ on $M$ for which the tangent bundle of $M$ is trivial as a complex vector bundle. Hence, the tangent bundle of $M$ supports a flat $J$-complex connection, i.e., a connection $A$ for which the complex structure is parallel and $F_A = 0$. A fortiori, we have $F^{0,2}_A = 0$, but $M$ does not admit a complex structure. Of course, the torsion of $A$ is non-zero and, in fact, must have a nontrivial $(0,2)$-piece.