  We introduce an intermediate family of Laurent polynomials between Schur's
$Q$-functions and S. Okada's symplectic $Q$-functions. It can also be regarded
as a $Q$-function analogue of Proctor's intermediate symplectic characters, and
is named the family of intermediate symplectic $Q$-functions. We also derive a
tableau-sum formula and a J\'ozefiak-Pragacz-type Pfaffian formula of the
Laurent polynomials.
