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No.
Cosmological redshift is a based on single universal frame of reference. The frame itself undergoes expansion, objects in the frame remain stationary (unless subject to gravity from neighbouring objects).
The time dilation for quasars is based on the distance of the quasar from the observer. The time dilation is calculated using the Lorentz transformation from special relativity.
Gravitational redshift is a direct consequence of the equivalence princple which says there is no difference between a stationary frame in a gravitational field and an accelerated frame moving in a gravity free field.
A photon moving in an accelerated frame will be observed to have a Doppler shift. In the stationary frame the photon will have the same Doppler (gravitational) shift if the strength of the field equals the acceleration of the frame in a gravity free field.
Regards
Steven
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