  In this paper we study the question of determining when an irreducible
admissible representation of ${\rm GL}_n(D)$ admits a symplectic model, that is
when such a representation has a linear functional invariant under ${\rm
Sp}_n(D)$, where $D$ is a quaternion division algebra over a non-Archimedian
local field $k$ and ${\rm Sp}_{n}(D)$ is the unique non-split inner form of the
symplectic group ${\rm Sp}_{2n}(k)$. We show that if a representation has a
symplectic model it is necessarily unique. For ${\rm GL}_2(D)$ we completely
classify those representations which have a symplectic model. Globally, we show
that if a discrete automorphic representation of ${\rm GL}_{n}(D_\mathbb{A})$
has a non-zero period for ${\rm Sp}_{n}(D_\mathbb{A})$, then its
Jacquet-Langlands lift also has a non-zero symplectic period. A somewhat
striking difference between distinction question for ${\rm GL}_{2n}(k)$, and
${\rm GL}_n(D)$(with respect to ${\rm Sp}_{2n}(k)$ and ${\rm Sp}_n(D)$ resp.)
is that there are supercuspidal representations of ${\rm GL}_n(D)$ which are
distinguished by ${\rm Sp}_n(D)$. The paper ends by formulating a general
question classifying all unitary distinguished representations of ${\rm
GL}_n(D)$, and proving a part of the local conjectures through a global
conjecture.
