  We give a description of the moment graph for Bott-Samelson varieties in
arbitrary Lie type. We use this, along with curve neighborhoods and explicit
moduli space computations, to compute a presentation for the small quantum
cohomology ring of a particular Bott-Samelson variety in Type $A$. We also show
the conjecture $\mathcal{O}$ of Galkin, Golyshev, and Iritani holds for that
Bott-Samelson variety.
