  We prove a quadratic sparse domination result for general non-integral square
functions $S$. That is, we prove an estimate of the form
  \begin{equation*}
  \int_{M} (S f)^{2} g \, \mathrm{d}\mu \le c \sum_{P \in \mathcal{S}}
  \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} \,
\mathrm{d}\mu\right)^{2/p_{0}}
  \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert
g\rvert^{q_{0}^*}\,\mathrm{d}\mu\right)^{1/q_{0}^*}
  \lvert P\rvert,
  \end{equation*} where $q_{0}^{*}$ is the H\"{o}lder conjugate of $q_{0}/2$,
$M$ is the underlying doubling space and $\mathcal{S}$ is a sparse collection
of cubes on $M$. Our result will cover both square functions associated with
divergence form elliptic operators and those associated with the
Laplace-Beltrami operator. This sparse domination allows us to derive optimal
norm estimates in the weighted space $L^{p}(w)$.
