This is probably closer in spirit to what you're looking for than what you've received in the comments. If $(V^{2m}, J, \omega, g)$ is Calabi-Yau (which for me means that $J$ is integrable, and the first Chern class $c_1(V) = 0$), then one can say much more. In this case there exists a holomorphic nowhere vanishing $(m,0)$ form $\Omega$, called the "holomorphic volume form." The form $\Omega$ is unique up to multiplication by a nowhere vanishing holomorphic function. We don't need to assume that $\Omega$ is parallel, but Yau's theorem does tell us (if $V$ is compact) that we can change the metric, keeping the Kaehler class $[\omega]$ unchanged, to make $\Omega$ parallel (and consequently also the new metric will be Ricci-flat.) But I have digressed.

In such a situation, if $L$ is Lagrangian, then it is well known that the restriction of $\Omega$ to $L$ is equal to $e^{i \theta} \mathrm{Vol_L}$, where $\mathrm{Vol}_L$ is the volume form of $L$ (with the induced metric) and $e^{i \theta}$ is the "phase" of the Lagrangian, where $\theta : L \to \mathbb R/ (2 \pi \mathbb Z)$ is a smooth, multivalued function on $L$. In addition, the mean curvature $H$ of $L$ in $V$ is given by $H = J \nabla \theta$. So the *minimal Lagrangian* submanifolds (vanishing mean curvature) correspond to those with *constant* phase function $\theta = \theta_0$. In this case, *if $L$ is compact*, then the volume of $L$ is given by

$\mathrm{Vol} (L) = \int_L \mathrm{Vol}_L = \int_L e^{-i \theta_0} \Omega = e^{- i \theta_0} [\Omega] \cdot [L],$

which is topological. (It looks complex, but it's actually real, because $[\Omega]$ is a class in $H^g(V, \mathbb C)$.) If you prefer, you can just replace $\Omega$ by $e^{- i \theta_0} \Omega$ to get rid of the phase factor.

Such "minimal Lagrangian" submanifolds, whose volume is purely topological, are also called *special Lagrangian* submanifolds, and are widely studied in calibrated geometry and differential geometric approaches to mirror symmetry. The best place to start looking is the text "Riemannian Holonomy Groups and Calibrated Geometry" by Dominic Joyce and its multiple references.

I'm not sure how much of this will extend to the case of $J$ non-integrable and $c_1(V) \neq 0$. I'd have to think about it.