  Given a triangulated region in the complex plane, a discrete vector field $Y$
assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector
field holomorphic if it defines an infinitesimal deformation of the
triangulation that preserves length cross ratios. We show that each holomorphic
vector field can be constructed based on a discrete harmonic function in the
sense of the cotan Laplacian. Moreover, to each holomorphic vector field we
associate in a M\"obius invariant fashion a certain holomorphic quadratic
differential. Here a quadratic differential is defined as an object that
assigns a purely imaginary number to each interior edge. Then we derive a
Weierstrass representation formula, which shows how a holomorphic quadratic
differential can be used to construct a discrete minimal surface with
prescribed Gau{\ss} map and prescribed Hopf differential.
