The long transmission distance of satellite ground quantum communication and the relative motion between the satellite and ground increase the tracking performance requirements of a quantum tracker. To improve the tracking accuracy of satellite ground quantum communication and reduce the error rate of quantum communication， the tracking and pointing control system of a quantum tracker is investigated in this study， and the total disturbance affecting the tracking accuracy is compensated. First， the satellite ground quantum communication link and the quantum communication process and its influencing factors are introduced. A mathematical model of the quantum tracker is then constructed. Based on the mathematical model of the tracker， an active disturbance rejection control （ADRC） model predictive control algorithm is designed and proposed. To solve the problem of mode switching， a fractional order tracking differentiator is proposed to optimize the trajectory and thus reduce overshoot. Experimental results of quantum communication with the “Mozi” satellite reveal that the fractional order tracking differentiator improves the target acquisition speed by 22% following mode switching. Compared with the traditional PI control， the proposed ADRC model predictive control has stronger anti-interference capabilities， weakens the miss distance peak of pitch reversal， and has a tracking accuracy that reaches 2.9". The quantum polarization data reception is improved， and the total bit error rate is reduced to 1.18%. The proposed control algorithm can further improve the tracking accuracy of the quantum tracker， reduce the bit error rate， and meet the accuracy requirements of satellite ground quantum communication.