In this study, we proposed a fractional order model of hepatitis B virus transmission dynamics with two-age structure under vaccination. A qualitative analysis of the model is performed. Basic reproduction number of the model is determined. Local stability conditions of disease-free equilibrium point are proven by using fractional Ruth-Hurwitz conditions. Global stability conditions of both disease-free and endemic equilibrium points are shown by constructing appropriate Lyapunov functions. Sensitivity analysis was done by using normalized forward sensitivity index approach. Numerical simulation is performed to investigate the effect of memory on hepatitis B disease dynamics by varying order of derivatives and to simulate the effects of vaccinating newborns immediately after birth, vaccinating children and adult vaccination. Then, we compared their effects on hepatitis B disease dynamics in the sense of control and elimination. It is observed that the number of infective individuals decreases faster and even falls to zero over a long run for the model with memory than memory-less model. Comparing results between vaccination of different ages show that increasing newborn vaccination immediately after birth has the highest effect on hepatitis B disease control.