Is it possible to write '$f$ is NOT a bijection' with quantifiers only, and without using "$\neg$"? What is the negation of "$\exists$!"?
Assume that the formal language only has $=$, $f$, variables, and quantifiers. Then there is no formula of the type requested in the question.
Suppose that $\phi$ is a formula that does not include $\lnot$. I claim that, in a structure with only one element, $\phi$ must be true. But every function from that model to itself is a bijection, so no $\phi$ can express "$f$ is not a bijection" in all models.
To prove that such a formula $\phi$ is true a model with only one element, we just have to consider all the cases, where we only consider formulas that do not include $\lnot$.
Every substitution instance of the formula $x = y$ will be true, because there is only one element. Every atomic formula is a substitution instance of $x=y$.
Every expression of the form $\psi \land \theta$, $\psi \lor \theta$, or $\psi \Rightarrow \theta$ will be true when $\psi$ and $\theta$ are both true
Every quantified expression $(\exists x)\psi$ of $(\forall x)\psi$ will be true, because every substitution instance of $\psi$ will be true.
Once we exclude models with only one element, the question has a postive answer. $(\forall x)(\forall y)(x = y)$ is then a suitable replacement for $\bot$. an identically false formula. And then $\lnot \theta$ is $\theta \Rightarrow \bot$ as usual, so we can express negation without using $\lnot$.