  We study a family of classical solutions of modified sinh-Gordon equation,
$\partial_z\partial_{{\bar z}} \eta-\re^{2\eta}+p(z)\,p({\bar z})\
\re^{-2\eta}=0$ with $p(z)=z^{2\alpha}-s^{2\alpha}$. We show that certain
connection coefficients for solutions of the associated linear problem coincide
with the $Q$-function of the quantum sine-Gordon $(\alpha>0)$ or sinh-Gordon
$(\alpha<-1)$ models.
