The description of the agile property reads:
Agile weapons are unusually well balanced and responsive. A wielder with the Weapon Finesse feat can choose to apply her Dexterity modifier to damage rolls with an agile weapon in place of her Strength modifier. This modifier to damage is not increased for two-handed weapons, but is still reduced for off-hand weapons.
Since the text here specifically says that the modifier to damage doesn't increase when using a two-handed weapon like it would when using your Str bonus, I would argue that the agile property is incompatible with Dragon Style in this regard.
On the other hand, this FAQ item clarifies that an Unchained Rogue using Finesse Training with a two-handed weapon gains 1.5 * Dex to damage. In that case, I think Dragon Style could be used to increase your damage bonus from Dex. So while I wouldn't expect to get away with it in Pathfinder Society play, I think you could reasonably rule to allow using Dragon Style to increase your Dex bonus to damage in your game specifically.
However you rule about using Dex for the damage portion of Dragon Style, though, I don't think that the monk can use his Dex modifier instead of his Str modifier to increase the duration of the shaken effect from Dragon Ferocity, since that isn't a form of damage.