  Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting
measure of particles of generation $n$. Under some conditions, we give a
probabilistic proof for the existence of the Yaglom limit of
$(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal
decomposition and the many-to-few formula. As a result, we give explicit
integral representations of all quasi-stationary distributions of
$(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and don't
rely on Martin boundary theory.
