= Solution
For degree zero, <compactness> and the <maximum modulus principle> make <holomorphic functions> constant on each <connected component>. For degree two, a <holomorphic differential form> is $\bar\partial$-closed and its $\partial$ derivative has type $(3,0)$, which vanishes on a complex surface. Degrees above two have no nonzero holomorphic forms. It remains to treat a <holomorphic differential form> $\alpha$ of degree one.
Set $\beta=d\alpha=\partial\alpha$. Its coefficients are holomorphic, so $\beta$ is a <holomorphic differential form> of degree two, hence closed by the degree-two observation. Its conjugate is closed too. The <Stokes theorem> gives
$$
\int_M\beta\wedge\bar\beta
=\int_Md(\alpha\wedge\bar\beta)=0.
$$
The integrand is nonnegative in the complex orientation. In local coordinates, if $\beta=b(z)\,dz_1\wedge dz_2$ with $z_j=x_j+iy_j$, then
$$
\beta\wedge\bar\beta
=4|b(z)|^2\,dx_1\wedge dy_1\wedge dx_2\wedge dy_2.
$$
It is strictly positive wherever $\beta$ is nonzero. Its zero integral therefore forces $\beta=0$ everywhere. Thus <holomorphic forms on a compact complex surface are closed>, even when the surface is not Kähler:
$$
\boxed{d\alpha=0\quad\text{for every global holomorphic }p\text{-form}.}
$$
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