Exists first-order formula $Z(x)$ in language $\langle 0, +, \le\rangle$ that over field $\mathbb{Q}$ expresses fact that $x$ is integer?

You are correct: no such formula exists.

Suppose that $Z(x)$ is such a formula. That is, for each $q \in \mathbb{Q}$ $$\mathfrak{Q} = \langle \mathbb{Q}, 0 , + , \leq \rangle \models Z[q] \Leftrightarrow q \in \mathbb{Z}. \tag{1}$$

Note that the mapping $\sigma (x) = \frac{x}{2}$ is an isomorphism of $\mathfrak{Q}$, and so for each $q \in \mathbb{Q}$ we have that $$\mathfrak{Q} \models Z[q] \Leftrightarrow \mathfrak{Q} \models Z[ \sigma(q) ]. \tag{2}$$ Now consider what happens when $q = 1$: By (2) we have that $$\mathfrak{Q} \models Z[1] \Leftrightarrow \mathfrak{Q} \models Z[\sigma(1) = \tfrac{1}{2}],$$ which clearly contradicts our assumption (1).