In an abelian category, I understand that $\text{Hom}_C(X,Y)$ forms an abelian group for all $X, Y \in \text{Ob}(C)$. Thus to show that $Q$ is an injective object implies $\text{Hom}_C(\cdot, Q)$ is an exact functor, we take the hom-sets to be objects in the category of abelian groups. Thus taking the $\text{Hom}_C(0, Q) = 0$ itself.
Thus assume $Q$ is an injective object, i.e. that for every monomorphism $f : X \hookrightarrow Y$ and morphism $g : X \to Q$ there is a morphism $h : Y \to Q$ such that $h \circ f = g$.
We want to show that the hom functor $\text{Hom}_C(\cdot, Q)$ applied to a short exact sequence in $C$:
$0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0$
yields another short exact sequence:
$$ 0 \leftarrow \text{Hom}_C(X, Q) \xleftarrow{- \circ f} \text{Hom}_C(Y, Q) \xleftarrow{-\circ g} \text{Hom}_C(Z, Q) \leftarrow 0 $$ where $- \circ f \equiv \text{Hom}_C(f, Q)$ is notation.
So to show that the first on the left is exact we need to show that $-\circ f$ is surjective. I.e. for all $g' : X \to Q$ there exists a hom $h : Y \to Q$ such that $(-\circ f) \circ h = g'$. But that is true by definition of injective.
Next is to show exactness of the abelian group on the far right. And that is the same is showing that $-\circ g$ is injective. That is if $g_1, g_2 \in \text{Hom}_C(Z, Q)$ then $g_1 \circ g = g_2 \circ g \implies g_1 = g_2$.
Not sure how to prove that since there's nothing about uniqueness in the definition of injective object and the existing morphism $h$.
You know that $g$ is an epimorphism, by the exactness of $0 \to X\to Y \xrightarrow{g} Z\to 0$. Hence the implication "if $g_1 \circ g=g_2 \circ g$ then $g_1=g_2$" is true for free, without using the injectivity of $Q$.
Indeed, for every object $Q$ it is the case that an exact sequence $X \to Y \to Z \to 0$ is sent by the functor $\text{Hom}(-,Q)$ to an exact sequence $\text{Hom}(X,Q) \leftarrow \text{Hom}(Y,Q) \leftarrow \text{Hom}(Z,Q)\leftarrow 0$, that is, $\text{Hom}(-,Q)$ is always left exact, even if $Q$ is not required to be injective. The injectivity of $Q$ is only used to prove that $\text{Hom}(-,Q)$ sends monomorphisms to surjections, as you did when you proved that $\text{Hom}(f,Q)$ is surjective by using that $f$ is monic and by the injectivity of $Q$.