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][1] 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][2] 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.


  [1]: http://www-math.mit.edu/~poonen/papers/subrings.pdf
  [2]: http://www.math.uu.nl/people/cornelis/publications/mazur's_conjecture.pdf