  The Fock space $\mathcal{F}(\mathbb{C}^n)$ is the space of holomorphic
functions on $\mathbb{C}^n$ that are square-integrable with respect to the
Gaussian measure on $\mathbb{C}^n$. This space plays an important role in
several subfields of analysis and representation theory. In particular, it has
for a long time been a model to study Toeplitz operators. Esmeral and Maximenko
showed in 2016 that radial Toeplitz operators on $\mathcal{F}(\mathbb{C})$
generate a commutative $C^*$-algebra which is isometrically isomorphic to the
$C^*$-algebra $C_{b,u}(\mathbb{N}_0,\rho_1)$. In this article, we extend the
result to $k$-quasi-radial symbols acting on the Fock space
$\mathcal{F}(\mathbb{C}^n)$. We calculate the spectra of the said Toeplitz
operators and show that the set of all eigenvalue functions is dense in the
$C^*$-algebra $C_{b,u}(\mathbb{N}_0^k,\rho_k)$ of bounded functions on
$\mathbb{N}_0^k$ which are uniformly continuous with respect to the square-root
metric. In fact, the $C^*$-algebra generated by Toeplitz operators with
quasi-radial symbols is $C_{b,u}(\mathbb{N}_0^k,\rho_k)$.
