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Yes. It's the square you are leaving that provokes, not the square you are entering. Since the square they are leaving is without cover, an AoO could theoretically be provoked.
That said, if it is their first turn there is a decent chance that they may win initiative, in which case the potential threat will be unable to utilize an AoO against them. Combat Reflexes and Uncanny Dodge would both allow AoOs despite this, in which case the monster will have to hope that the opponent has not yet drawn their weapons (which they likely have not if they have not yet acted).
TL;DR - Burrowing could provoke, but there are a variety of more mundane ways they can get out of the AoO.
Does a monster w/ a burrowing speed prevoke an AoO if they burrow stright down in the square that they are in at the begining of thier turn?
I would say so, unless they 5 ft stepped strait down. Combat is technically 3 dimensional. You provoke attacks by flying through a threatened space, burrowing would be the same. The catch is burrowing is a pain since you might not be able to make the AoO due to the cover/concealment granted by the ground.
As I understand it, the AoO occurs in the square being left, so the ground doesn't feature into the calculation. Though it would for an AoO triggered by a burrowing creature that was already underground.
As I understand it, the AoO occurs in the square being left, so the ground doesn't feature into the calculation. Though it would for an AoO triggered by a burrowing creature that was already underground.
That's true, but with even marginal effort it won't matter. If you use a withdraw action you don't provoke from the first space you move from, and the rest of those spaces provide you cover/concealment. It's pretty hard to stop a burrower once they get going.
A really awesome encounter I once ran was a group of giant centipedes who had their climb speeds replaced by burrow speeds (making them something like graboids from Tremors). They would burst up out of the ground, bite people, and so forth. Sometimes they would try to grapple with you and pull you into the ground with them (grapple move, see grapple), which was almost assuredly doom. The party was very clever about avoiding them, and tried to not get cornered by any of them.