This paper provides a comprehensive analysis of almost maximum distance separable (AMDS) constacyclic codes of length $4p^ς$ over the finite field $\mathbb{F}_{p^m}$, where $p$ is an odd prime number. Furthermore, it introduces the construction of quantum AMDS (qAMDS) codes, drawing on the principles of the Calderbank-Shor-Steane (CSS) framework, which enhances their applicability in quantum error correction. This work aims to deepen the understanding of these codes and their potential uses in both classical and quantum computing environments.