I'm a little new to this stuff so I might not know exactly what you're asking.
I believe there is a conjecture of Mazur that implies the type of description of $\Bbb Z$ you are looking for is impossible.
This paper by Poonen would be a good place to start. Of course Poonen would be able to give a much more satisfying answer. I only caught the end of his talk about this stuff at the Joint Meetings.
EDIT:
Here Cornelissen and Zahidi show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of $\Bbb Z$ over $\Bbb Q$. They also show that the analogue of Mazur's conjecture is false in the function field case, where Hilbert's Tenth has a negative answer.