There isn’t a whole lot of information on Dragon Quest III HD-2D Remake yet, even though the game had been announced by Square Enix earlier this year. There isn’t a release date at all, and even the platforms it will be releasing on isn’t really known yet.
However, there might be an end to the silence soon. In an interview with Famitsu that Nintendo Everything posted a translation of, Dragon Quest creator Yuji Horii mentioned that “I can’t say much yet, but we might be able to deliver more news on DQIII soon”.