  We realize a family of generalized cluster algebras as Caldero-Chapoton
algebras of quivers with relations. Each member of this family arises from an
unpunctured polygon with one orbifold point of order 3, and is realized as a
Caldero-Chapoton algebra of a quiver with relations naturally associated to any
triangulation of the alluded polygon. The realization is done by defining for
every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the
referred quiver with relations, and by proving that for every triangulation
$\tau$ and every arc $j\in\tau$, the product of the Caldero-Chapoton functions
of $M(j)$ and $M(j')$, where $j'$ is the arc that replaces $j$ when we flip $j$
in $\tau$, equals the corresponding exchange polynomial of Chekhov-Shapiro in
the generalized cluster algebra. Furthermore, we show that there is a bijection
between the set of generalized cluster variables and the isomorphism classes of
$E$-rigid indecomposable decorated representations of $\Lambda$.
