This is an interesting question! Suppose that the object we are considering is spherical, and first let’s consider that the light source is coming from the center of the object. If we assume that the medium of the object does not interfere with the light as it passes through the object, then we would not observe any red shifting via the Doppler Effect. We would observe some sort of gravitational red shift.

However, if the light source were at the surface of the object, then we would see some type of red shifting. This is because the [front] surface is moving away from us due to its contraction, and hence it would be red shifted via the Doppler Effect.

Interestingly, if the object were transparent to its own light source, then you would observe a blue-shift of light from the other side.

Ultimately, the red shifting is likely to be negligible because the object would need to contract/implode at an immense rate in order to have sufficient velocity for the Doppler Effect.