Stretching is always the best when you are warm, so after walks, workouts, or even hot showers or baths. Avoid doing stretches when body is cold as there is more likelyhood of tearing muscles.
Jen
The only problem I could see with stretching in the morning is that the discs in your spine fill with fluid while you sleep making them more prone to injury for about 30 mins to an hour after you wake.
If I were to stretch in the morning I would probably get up, have breakfast, take shower, and then stretch.
I don't know a ton about Pilates, but as long as you are not doing much dynmaic stretching (which isn't all that great to begin with), particularly with stretches involve your spine, you should be alright to stretch in the morning.
There would be more cause for concern if you were waking up and immediately going to do heavy deadlifts, squats, or olympic lifts since those are very taxing on the CNS and involve a lot of spinal stability.