It appears that Square Enix will be making a new Dragon Quest themed announcement in June this year. The news comes to us from a forum post made by a user who claims to have seen the next issue of Weekly Shonen Jump, which includes an announcement about the Dragon Quest series by SE. As the next issue of Weekly Shonen Jump is scheduled to release on June 10th, it coincides nicely with E3. And Square is hosting their own E3 showcase again this year. So obviously, the announcement will likely happen then.
Here's a loose translation thanks to Dualshockers:
The next Weekly Shonen Jump issue has Square Enix teasing news regarding the Dragon Quest series, leaks revealed. Square Enix teased that this year's 28th issue of Weekly Shonen Jump, releasing on June 10, will have an announcement regarding the Dragon Quest series.
June 10 coincides with the Square Enix E3 conference, so it's highly likely a new Dragon Quest game will be announced then.
Dragon Quest Director Yuji Horii recently stated that Dragon Quest XII is "many years ahead", meaning this can't be the reveal of a new main Dragon Quest game.
So while this is very likely not Dragon Quest XII, it could be DLC for Dragon Quest XI, a remake of an older Dragon Quest game, or an announcement concerning the Dragon Quest Monsters game that is currently in development. It probably isn't a Dragon Quest X announcement, as that game just received a major content update a few days back.
We'll have to wait until June 10th to find out, but for now, enjoy speculating just what exactly the announcement could be.