Let $X$ be a compact Riemann surface, and let $q \in K^{\otimes2}(X)$ be a holomorphic quadratic differential
on $X$. Let $\Lambda_{q}$ be the sheaf of holomorphic vector fields $\chi$ satisfying
$\mathcal{L}_\chi q = 0$, where $\mathcal{L}$ is the Lie derivative. On p. 265-266 of [Quadratic differentials and foliations](https://projecteuclid.org/euclid.acta/1485890020) (Acta **142** (1979), 221-274), Hubbard and Masur use the following fact: 

If $q$ is the square of a holomorphic one-form, then 
$H^2(X, \Lambda_q) = \mathbb{C}$. Otherwise,  $H^2(X, \Lambda_q)=0$.

How does one prove this fact?

More information:
$X\setminus q^{-1}(0)$ is covered by *flat charts* for $q$, i.e. holomorphic charts $(U,z)$ with respect to which
$q=(dz)^2$. Transitions between these charts are of form $z\mapsto \pm z+c$ with $c\in \mathbb{C}$.
If $(U,z)$ is a flat chart, then $\Lambda_q(U)$ is the one-dimensional space of constant vector fields $\left\{ a \frac{d}{dz} \middle| a\in \mathbb{C}  \right\}$.
(On the other hand, if $U$ contains a zero of $q$, $\Lambda_q(U)=0$.)
The differential $q$ is a square if and only if the bundle of horizontal tangent vectors
$\left\{ a \frac{d}{dz} \middle| a\in \mathbb{R}  \right\}$ over $X\setminus q^{-1}(0)$ is orientable,
if and only if there is a cover of $X\setminus q^{-1}(0)$ by flat charts so that the transition
functions are all translations: $z\mapsto z+c$.