This trophy is called "Come Rain or Shine" and it's awarded for experiencing every type of weather in every wild zone.
There are a total of six Wild Zones within Dragon Quest Heroes II in which you must experience a change in weather. To experience a weather change, all you need to do is spend several minutes in the Wild Zone. There's the initial weather when entering the zone, and then only one change while you're within the zone.
Here's the zone and the weather in each of them:
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Greena Pastures - Sun, Rain
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The Grand Dunes - Sun, Sandstorm
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The Lumberlands - Sun, Fog
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No Man's Land - Overcast, Fog
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Redrock Ravine - Sun, Rain
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The Frosty Foothills - Overcast, Blizzard
If you do each zone in order, then once the Blizzard begins in the Frosty Foothills, the trophy will unlock: