Is there a first order formula $\varphi[x]$ in $(\mathbb Q, +, \cdot, 0)$ such that $x≥0$ iff $\varphi[x]$?

Yes. Every rational number $\ge 0$ is the sum of four squares. This is easily derived from Lagrange's Theorem which says that every non-negative integer is the sum of four squares.

Note that the formula we get is existential.

It is a famous result of Julia Robinson that $(\Bbb{Q}, +, \cdot, 0)$ is undecidable. This implies that $(\Bbb{Q}, +, \cdot, 0)$ does not admit elimination of quantifiers. That the rational numbers are not definable in the first-order theory of the reals follows from this, but also follows from well-known facts about O-minimality of the first-order theory of the reals