By definition a $\mathbb{P}$ - generic filter $G$ over a ground model $M$ is $\aleph_0$ - complete because for any finite set of conditions in $G$ there is a condition $p\in G$ such that $p$ is stronger than them all. This is similar to being closed under intersection of finitely many members for a usual filter on a set.
Q) What are examples of sufficient conditions on forcing notion $\mathbb{P}$ and possibly ground model $M$, such that any $\mathbb{P}$ - generic filter $G$ over $M$ is $\kappa$ - complete in the sense that "for any set of conditions of size $<\kappa$ in $G$ there is a condition $p\in G$ ( not $p\in \mathbb{P}$) such that $p$ is stronger than them all."