Can red shift and blue shift exist on the same object at the same time?

Yes, if the object is spinning there will be red shifts on the half moving away and blue shifts on the half moving closer.

This is one way tornadoes are detected using Doppler RADAR, the water droplets on one side of the cloud are moving away (red shift) while those on the other side are moving closer (blue shift), showing that the cloud is rotating.

Solid spinning objects act the same and the rotation is detectable with Doppler RADAR.