I am going to interpret "the Weil conjectures" as a synecdoche for "thinking about counting points in terms of eigenvalues of Frobenius operators." In this case, the answer is a resounding yes! However, the specific facts about eigenvalues of Frobenius which which are being used here are easier than the ones discussed in the Weil conjectures. Silverman's *The Arithmetic of Elliptic Curves* might be a good reference for you.

Let $E$ be the genus one curve $X^3+Y^3+Z^3$ over $\mathbb{F}_p$. This curve has an automorphism $\Phi: (x:y:z) \mapsto (z^p:y^p:z^p)$. The map $\Phi$ sends every point with values in $\mathbb{F}_p$ to itself. However, points with values in $\mathbb{F}_{p^n}$ for higher values of $n$ are permuted. Our goal is to find the number of fixed points of $\Phi$.

The Weil conjecture way to do this is to associate to $E$ certain vector spaces $H^0(E)$, $H^1(E)$ and $H^2(E)$. The map $\Phi$ acts functorially on cohomology, and we have
$$\#(\mbox{Fixed points of }\Phi) = Tr(\Phi^*:H^0 \to H^0) - Tr(\Phi^*:H^1 \to H^1) + Tr(\Phi^*:H^2 \to H^2).$$
Now, in this case, you can really get quite explicit about what these vector spaces are. $H^0$ and $H^2$ are both $1$-dimensional, and $\Phi$ acts by $1$ and $p$ respectively. $H^1$ the Tate module of $E$ (Silverman III.7); a two dimensional space.

So the number of fixed points is $p+1-a$ where $a$ is the trace of $\Phi^*$ acting on the Tate module.

One can show that the endomorphism ring of $E$ is $\mathbb{Z}[\omega]$ where $\omega$ is a primitive cube root of unity (see Silverman V.3). So we must have $\Phi = c+d \omega$ for some integers $c$ and $d$. The fact that $\Phi$ has degree $p$ (see Silverman II.2) means that $N(c \omega +d) = c^2 - cd + d^2 =p$. And one computes that $a = Tr(\Phi) = 2c - d$.

So, in short, the number of points on $E$ is $p+1-a$, where $a=2c-d$ with $(c,d)$ such that $N(c+d \omega)=p$. Converting that into your statement, and getting the signs right, is left to you.

Focusing on the more specific question of using the Weil conjectures. The fact that $N(c+d \omega)$ is $p$ comes from the Riemann hypothesis part of the Weil conjectures. But you also need to know that $\mathbb{Z}[\omega]$ is the full ring of automorphisms, which is not particularly related to the Weil conjectures. And you need to figure out which of the $6$ elements of $\mathbb{Z}[\omega]$ which have norm $p$ is the correct one, which is also not a Weil computation. So I would say that this is not specifically a Weil conjecture result, although definitely can be proven using the same sort of ideas as in the Weil conjectures.