Yes, this ought to be true. One way to prove this would be to produce an equivalence to the universal cocartesian fibration defined in Lurie's Kerodon. Here is an outline of such a proof.
N.B. I have't yet worked out all the details in steps 3 and 4 below, so read this answer with a grain of salt.
The proof proceeds in four steps:
We first define a Kan-enriched category $\mathbf{Rep}$, and show that its homotopy coherent nerve is equivalent (over the quasi-category of quasi-categories) to the quasi-category described in your question.
We next show that the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ is sent by the homotopy coherent nerve functor to a cocartesian fibration.
We next define a morphism of cocartesian fibrations from the homotopy coherent nerve of $\mathbf{Rep} \to \mathbf{qCat}$ to the universal cocartesian fibration defined in Lurie's Kerodon.
Finally, we show that this morphism of cocartesian fibrations is an equivalence on fibres, and hence an equivalence of cocartesian fibrations.
1. We begin by defining the Kan-enriched category $\mathbf{Rep}$. Its objects are pairs $(A,a)$, where $A$ is a quasi-category and $a$ is an object of $A$. We define a morphism from such an object $(A,a)$ to another $(B,b)$ to be a commutative square of quasi-categories
$\require{AMScd}$
\begin{CD}
A/a @>>> B/b \\
@VVV @VVV\\
A @>>> B
\end{CD}
where the vertical maps are the projections. Moreover, we define the hom Kan complexes of $\mathbf{Rep}$ by the following pullback squares of Kan complexes, where $\mathbf{qCat}$ denotes the usual Kan-enriched category of quasi-categories.
\begin{CD}
\mathrm{Hom}_{\mathbf{Rep}}((A,a),(B,b)) @>>> \mathrm{Hom}_\mathbf{qCat}(A/a,B/b)\\
@V V V @VV V\\
\mathrm{Hom}_\mathbf{qCat}(A,B) @>>> \mathrm{Hom}_\mathbf{qCat}(A/a,B)
\end{CD}
Composition in $\mathbf{Rep}$ is defined in terms of the composition in $\mathbf{qCat}$.
Observe that the left-hand vertical map in the above pullback square is a Kan fibration, since the right-hand vertical map is. These left-hand vertical maps define the action on homs of a simplicial functor $\mathbf{Rep} \to \mathbf{qCat}$, which is given on objects by $(A,a) \mapsto A$.
It follows from HTT.4.2.4.4 that a certain explicit functor from the homotopy coherent nerve of $\mathbf{Rep}$ to the quasi-category you define in your question is an equivalence over $N(\mathbf{qCat})$. (I would be happy to explain this in more detail, but for now let me press on).
2. We now show, using HTT.2.4.1.10, that the homotopy coherent nerve functor $N$ sends the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ to a cocartesian fibration. Since this forgetful functor is a Kan fibration on homs, it is sent by $N$ to an inner fibration.
Given an object $(A,a)$ of $\mathbf{Rep}$ and a morphism $F \colon A \to B$ of $\mathbf{qCat}$, there is a lift $(A,a) \to (B,Fa)$ in $\mathbf{Rep}$ given by the commutative square whose bottom map is $F \colon A \to B$, and whose top map is the functor $F/a \colon A/a \to B/Fa$ induced by $F$. To show that this morphism is cocartesian with respect to the forgetful functor, we must show that, for any object $(C,c)$ of $\mathbf{Rep}$, the commutative square of Kan complexes
$\require{AMScd}$
\begin{CD}
\mathrm{Hom}_{\mathbf{Rep}}((B,Fa),(C,c)) @>(F/a)^*>> \mathrm{Hom}_\mathbf{Rep}((A,a),(C,c))\\
@V V V @VV V\\
\mathrm{Hom}_\mathbf{qCat}(B,C) @>>F^*> \mathrm{Hom}_\mathbf{qCat}(A,C)
\end{CD}
is a homotopy pullback square. Since the vertical maps are Kan fibrations, it suffices to check this on fibres, i.e., that for every functor $G \colon B\to C$, the restriction map $$\mathrm{Hom}_B(B/Fa,G^*(C/c)) \to \mathrm{Hom}_A(A/a,(GF)^*(C/c))$$ is an equivalence of Kan complexes. But this follows from the Yoneda lemma, which shows that this map is equivalent to the identity on the right-hom space $\mathrm{Hom}_C^R(GFa,c)$. (Again, I can explain this in more detail, if you wish.)
Hence the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ is sent by the homotopy coherent nerve functor to a cocartesian fibration.
3. In Kerodon, Lurie constructs a universal cocartesian fibration, which he denotes by $\mathcal{Q}\mathcal{C}_*^\mathrm{lax} \to \mathcal{Q}\mathcal{C}$. Here $\mathcal{Q}\mathcal{C}$ is the homotopy coherent nerve of the Kan-enriched category $\mathbf{qCat}$, and $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is defined as the "pith" of a certain ∞-bicategory. (See https://kerodon.net/tag/020S .) Explicitly, an $n$-simplex of $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is a simplicial functor $\mathfrak{C}(\Delta^{n+1}) \to \mathbf{sSet}$ whose value at $0$ is $\Delta^0$ and whose restriction to $\mathfrak{C}(\Delta^{\{1,\ldots,n+1\}})$ factors through $\mathbf{qCat}$. The universal cocartesian fibration sends such an $n$-simplex to the $n$-simplex of $\mathcal{Q}\mathcal{C}$ given by that restricted simplicial functor.
Thus an object of $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is the same thing as an object of $\mathbf{Rep}$, but a morphism $(A,a) \to (B,b)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is a functor $F \colon A \to B$ together with a morphism $g \colon Fa \to b$ in $B$. Such a morphism is cocartesian iff $g$ is an isomorphism in $B$.
Now comes the hard part, which is to define an explicit morphism of cocartesian fibrations over $N(\mathbf{qCat})$ from $N(\mathbf{Rep})$ to $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$. One can begin to construct such a morphism, call it $T$, in low dimensions by hand, but I haven't yet worked out how to give an explicit combinatorial description in all dimensions. On $0$-simplices, $T$ is the identity, i.e. sends $(A,a)$ to $(A,a)$. On $1$-simplices, $T$ sends a morphism $(F,\widetilde{F}) \colon (A,a) \to (B,b)$ in $N(\mathbf{Rep})$, i.e. a commutative square
\begin{CD}
A/a @>\widetilde{F}>> B/b \\
@VVV @VVV\\
A @>>F> B
\end{CD}
to the morphism $(F,\widetilde{F}(a)) \colon (A,a) \to (B,b)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ whose second component $\widetilde{F}(a) \colon Fa \to b$ is the image under $\widetilde{F}$ of the final object of $A/a$. One can continue to define $T$ on $2$-simplices, but it becomes more involved, and I haven't yet figured out to describe $T$ for all $n$-simplices.
To see that $T$ preserves cocartesian morphisms, it suffices to observe that it sends the cocartesian lift $(F,F/a) \colon (A,a) \to (B,Fa)$ in $N(\mathbf{Rep})$ described in section 2 above to the morphism $(A,a) \to (B,Fa)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ given by the functor $F \colon A \to B$ and the identity morphism on $Fa$ in $B$, which is cocartesian by what was said above.
4. It remains to show that the morphism of cocartesian fibrations $T$ (not yet fully) defined in section 3 is an equivalence on fibres. Let $A$ be a quasi-category. The fibre of the cocartesian fibration $N(\mathbf{Rep}) \to N(\mathbf{qCat})$ above $A$ is $N(\mathbf{Rep}_A)$, i.e. the homotopy coherent nerve of the full subcategory of $\mathbf{sSet}/A$ spanned by the projections $A/a \to A$, for $a \in A$. Whereas the fibre of Lurie's universal cocartesian fibration over $A$ is the left-hom quasi-category $\mathrm{Hom}^L_{N(\underline{\mathbf{qCat}})}(\Delta^0,A)$. (Here $\underline{\mathbf{qCat}}$ denotes the quasi-category enriched category of quasi-categories.) Both of these fibres are equivalent to $A$. Moreover, the induced map between these fibres is bijective on objects, and it will hopefully not be too much work to show that it is also an equivalence on homs, once the map $T$ above has been fully defined.