There's perhaps some relationship, and of course "martial law" can have a wide variety of definitions under different political contexts, but I think it's worth pointing out that the Roman Republic worked pretty differently to most modern democracies despite some nominal similarities. Most of the executive offices of the Republic were extraordinarily weak by modern standards (and had more checks on them than their modern equivalents). This extended to the dictatorship which was really more akin to appointing a temporary executive with the power to run a single part of the government with fewer (but still some!) checks on their authority for a short period of time.