symmetric monoidal (∞,1)-category of spectra
Ring is the category of rings (with unit) and ring homomorphisms (that preserve the unit).
A ring is a monoid in Ab, where $Ab$ is the category of abelian groups. So, $Ring$ is an example of a category of internal monoids.
Every surjective homomorphism of rings is an epimorphism in $Ring$, but not every epimorphism is surjective.
A counterexample is the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of integers into the ring of rational numbers. This is an injective epimorphism of rings.
For more see for instance at Stacks Project, 10.106 Epimorphisms of rings.