There should be some general proof. Of course, proper boundary condition is needed. Here, the manifold is a general Riemann manifold.

Can anyone give one?


You just need the integration by parts formula $$ \int \nabla f \cdot X=-\int f \operatorname{div} X,$$ which holds for any vector field $X$ and function $f$ such that $fX=0$ on the boundary of your manifold. (You can derive this by applying the divergence theorem to the vector field $f X$.)

Thus for any $f,g$ vanishing on the boundary we have $$\langle f, \Delta g \rangle = \int f\ \overline{\operatorname{div}(\nabla g)} = -\int \nabla f \cdot \overline{\nabla g} = \int \operatorname{div}(\nabla f) \ \bar g=\langle \Delta f, g\rangle.$$


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