In this paper we introduce the concepts of prime and semiprime quasi-ideals in ternary semirings and give its characterization. Furthermore, we show that a quasi-ideal $Q$ of a ternary semiring $S$ is $S$-prime if and only if $RML\subseteq Q$ implies $R\subseteq Q$ or $M\subseteq Q$ or $L\subseteq Q$ for any right ideal $R$, lateral ideal $M$ and left ideal $L$ of $S$.