Does the story of Dragon Quest continue after the anime?
According to Wikipedia (with a source):

The series adapts the events of the first 10 volumes of the manga, with initial plans to continue onward until scheduling and time slot changes at TBS lead to the series ending after 46 episodes. To accommodate the abrupt ending, [Riku] Sanjo [the manga's author] helped to provide an adjusted finale to the anime.

The manga apparently has 37 volumes, so yes, the story continues for a considerable amount past the point where the anime stopped.