We study a family of classical solutions of modified sinh-Gordon equation,
$\partial_z\partial_{{\bar z}} \eta-\re^{2\eta}+p(z)\,p({\bar z})\
\re^{-2\eta}=0withp(z)=z^{2\alpha}-s^{2\alpha}.WeshowthatcertainconnectioncoefficientsforsolutionsoftheassociatedlinearproblemcoincidewiththeQ−functionofthequantumsine−Gordon(\alpha>0)orsinh−Gordon(\alpha<-1)$ models.Comment: 35 pages, 3 figure