It seems to me that because animals usually don't have much in the way of conscious volition they don't cause themselves to need any relaxation; they just need to rest - which is a different thing. Relaxation is more than just resting - for example metabolism can be reduced by up to 64% in meditation (as shown here), whereas in sleep it reduces around 10-15%. But I think that relaxation is something animals simply can't do, and have no need to do anyway.
Relaxation (as opposed to just resting) is really a conscious volitional activity to reduce other mental or physical activity which has been consciously originated. Animals just don't need it (unless perhaps they are reacting to their owner's stress).
It is with conscious volitional activity that we create stress in the first place, animals don't do this. Just like young children, when animals rest they just rest completely. They seem to be very good at relaxing, but that's because they have no relaxation need in the first place, and no stress to reduce (imagine a baby needing relaxation and you'll see what I mean).
I think this is exactly why animals (or young children) can help us find deep relaxation ourselves. Looking at it in terms of a quality like stillness, it would be like learning stillness from hanging out with a large rock.