There exists the following result in the literature: There exists a polarized $K3$ surface $(X, H)$ of genus $3$ and a smooth irreducible curve $C$ on $X$ satisfying $C^2 =4$, $C.H=6$ such that $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[C]$. The theorem follows from [https://arxiv.org/pdf/math/9805140.pdf] theorem $1.1(iv)$.

Now Let's consider $X$ to be a smooth quintic hypersurface in $\mathbb P^3$ with Picard number $2$. Then I think it can be shown that $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[H']$, where $H$ is the hyperplane class and $H'$ is some divisor. Now in order to locate the ample line bundles in $\text{Pic}(X)$ using the Nakai-Moishezon criterion, we must know the intersection numbers $H.H'$ and $H'^2$.

In this context my question is the following: Does there exist in the literature an analogous existence result as the first-mentioned theorem for smooth quintic hypersurface with Picard number $2$?

To be more precise: Does there exist a polarized smooth quintic hypersurface $(X, H)$ in $\mathbb P^3$ and a smooth irreducible curve $C$ for which the intersection numbers $C^2$ and $C.H$ are known and $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[C]$?

Can someone give me any reference which could be even remotely useful in the context of finding out such $(X,H)$ and $C$

Any help from anyone is welcome.