Accurate Q (quality factor) structures can provide important constraints for characterizing subsurface hydrocarbon/water resources in exploration geophysics and interpreting tectonic evolution of the Earth in earthquake seismology. The attenuation effects on seismic amplitudes and phases can be included in forward and inverse modeling by invoking a generalized standard linear solid rheology. Compared to traditional ray-based methods, full-waveform adjoint tomography, which is based on numerical solutions of the visco-elastodynamic wave equation, has the potential to provide more accurate Q models. However, applications of adjoint Q tomography are impeded by the computational complexity of Q sensitivity kernels, and by strong velocity-Q trade-offs. In this study, following the adjoint-state method, we show that the Q (P and S wave quality factors Q_P and Q_S) sensitivity kernels can be constructed efficiently with adjoint memory strain variables. A novel central-frequency difference misfit function is designed to reduce the trade-off artifacts for adjoint Q tomography. Compared to traditional waveform-difference misfit function, this misfit function is less sensitive to velocity variations, and thus is expected to produce fewer trade-off uncertainties. The multiparameter Hessian-vector products are calculated to quantify the resolving abilities of different misfit functions. Comparative synthetic examples are given to verify the advantages of this new misfit function for adjoint Q_P and Q_S tomography. We end with a 3D viscoelastic inversion example designed to simulate a distributed acoustic sensing/vertical seismic profile survey for monitoring of CO₂ sequestration.