For analytic functions we study the remainder terms of Gauss quadrature rules with respect to Bernstein-Szegő weight functionsw(t)=wα,β,δ(t)=1+t1−tβ(β−2α)t2+2δ(β−2α)t+α2+δ2,t∈(−1,1),where 0 < 𝛼 < 𝛽, ...