We consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures T(h) and T(c) (<T(h)). Although the behavior of spin-1/2 system differs substantially from that of the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression η(mp)≤η(+)≡η(C)(2)/[η(C)-(1-η(C))ln(1-η(C))] with η(C)=1-T(c)/T(h) as the Carnot efficiency. This expression η(mp) possesses the same universality of the CA efficiency η(CA)=1-√(1-η(C)) at small relative temperature difference. Within the context of irreversible thermodynamics, we calculate the Onsager coefficients and show that the value of η(CA) is indeed the upper bound of EMP for an Otto engine working in the linear-response regime.