How I can negate this case How I can negate this case: For all $q≥2$, there exist $n,v∈ℕ^{∗}$ and there exist $ξ,θ∈ℝ$ such that $q=f(n,v,ξ,θ)$ where $f$ is a known real function.

You have to negate each part of the sentence, using the fact that the negation of $\exists$ is $\forall$, and vice-versa.

There exists $q\geqslant 2$ such that for every $n,v \in \mathbb{N}^*$ and for every $\xi, \theta \in \mathbb{R}$, we have $f(n,v,\xi,\theta) \
eq q$.

(edit : I assumed $f$ is a known function, otherwise you'll have to add "and for every $f$" before the conclusion.)