In a paper by Voisin[1] she considers as an example a K3-surface whose Picard group is generated by the class of an elliptic curve.
... l'exemple d'une surface K3 dont le groupe de Picard est engendré par la classe d'une courbe elliptique ...
What would be a concrete example of such a K3 surface?
A surface $S \subset \mathbb P^1 \times \mathbb P^2$ of bidegree $(2,3)$ gives an elliptic fibration $S \to \mathbb P^1$, so there are plenty elliptic curves, all linearly equivalent, but according to [2, Example 5.8] the Picard group still has rank $2$.
[1] Claire Voisin, Surt la stabilité des sous-variétés lagrangiennes.
[2] Bert van Geemen, Some remarks on Brauer groups of K3 surfaces