  We prove that the set of CM points on the Shimura curve associated to an
Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in
bijection with the set of certain classes of $p$-adic binary quadratic forms,
where $p$ is a prime dividing the discriminant of the quaternion algebra. The
classes of $p$-adic binary quadratic forms are obtain by the action of a
discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising
from the $p$-adic uniformization of the Shimura curve. We finally compute
families of $p$-adic binary quadratic forms associated to an infinite family of
Shimura curves studied in a previous paper of Amor\'os-Milione. This extends
results of Alsina-Bayer to the $p$-adic context.
