By MDPR, c.e. sets are the same as Diophantine sets. An existential definition of $\Bbb Z$ in $\Bbb Q$ would solve Hilbert's 10th problem over $\Bbb Q$.
The idea is not so crazy, you just input a pair $(a,b)$ of integers (I guess a pair of pairs of naturals, but that's a detail) and if $b \vert a$ you halt and if it doesn't you run forever. So the pairs $(a,b)$ for which the algorithm halts are rational numbers that are actually integers and if it doesn't they're rational numbers that are not integers.
N.B. This is so simple I know I cannot be correct. Many people much more experienced than me have tried and failed with many more complicated methods. This is more of a request to explain why this is wrong.
What I am guessing could be my incorrect assumption:
- You can just give an algorithm that distinguishes between integers and rationals and by MDPR the c.e. set this algorithm produces is a Diophantine set and hence a Diophantine definition of $\Bbb Z$ in $\Bbb Q$.
- You can sneak the integers into my algorithm by defining rationals as pairs of integers and not worry about circularity.