Let's work with the curvature tensor:

Since $J$ is parallel, it holds that $R(u,v)Jw=JR(u,v)w$, and hence $R(u,v,Jw,Jx)=R(u,v,w,x)$. For this equality and for the symmetries of the curvature tensor, it happens that when describing $R$ in local coordinates, the only terms to be computed are $R_{j\bar{k}l\bar{m}}=R(\partial_{j},\partial_{\overline{k}},\partial_{l},\partial_{\overline{m}})$. Now

and also

In general,

so $\dfrac{\partial g^{ab}}{\partial z^c}=-g^{ad}g^{eb}\dfrac{\partial g_{ed}}{\partial z^c}$. Therefore,

and finally