  We present systematic measurements of the quality factors of surface acoustic
wave (SAW) resonators on ST-X quartz in the gigahertz range at a temperature of
$10 \, \textrm{mK}$. We demonstrate a internal quality factor $Q_\mathrm{i}$
approaching $0.5$ million at $0.5 \, \textrm{GHz}$ and show that
$Q_\mathrm{i}\geq4.0\times10^4$ is achievable up to $4.4 \, \textrm{GHz}$. We
show evidence for a polynomial dependence of propagation loss on frequency, as
well as a weak drive power dependence of $Q_\mathrm{i}$ that saturates at low
power, the latter being consistent with coupling to a bath of two-level
systems. Our results indicate that SAW resonators are promising devices for
integration with superconducting quantum circuits.
