It depends on the hazard in question.
From the combat section
You can’t move diagonally past a corner (even by taking a 5-foot step). You can move diagonally past a creature, even an opponent.You can also move diagonally past other impassable obstacles, such as pits.
I can't find anything that definitively states that pit traps are only triggered if you step in the square it occupies, but that's how I've seen them run. The rules state you can move diagonally past a pit, suggesting that this won't trigger it.
So,
*If the hazard in question is an impassable wall like structure then the samurai would not be allowed to make the diagonal movement. They would need to move at right angles around it.
*If the hazard in question is not an impassable wall like structure and is triggered by someone moving adjacent to it, they would trigger it as normal, moving in the way shown.
*If the hazard in question is not an impassable wall like structure and is triggered by someone entering one of the squares it occupies, then it is not triggered as the character technically did not enter a square occupied by the hazard. In the same way they wouldn't provoke an AoO from a creature that only threatens the squares occupied by the hazard.
It's a shame hex maps aren't easier to draw on. It makes situations like this a non-issue.
Venture-Lieutenant, Spain—Mirande de Ebro
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Hazard: The root cellar of the Sauer manor once extended away from the house with well-maintained beams keeping the ceiling supported. These have since collapsed, and a sinkhole is about to form. A character can notice the budding sinkhole in front of the house with a successful DC 13 Perception or Survival check. Anyone who steps in one of the hazard squares denoted on the map collapses the entire sinkhole and must succeed at a DC 20 Reflex save or slide into a 20-foot-deep pit, taking 1d6 points of damage, 1d6 points of nonlethal damage, and landing mired in soil (giving the victim the entangled condition). A character can extricate himself from the soil as a standard action and can climb out of the sinkhole with a successful DC 15 Climb check. Once the sinkhole has caved in, the 12 squares adjacent to the hazard area become slanted and are treacherous to stand near. Anyone ending his turn in one of those squares must succeed at a DC 15 Reflex save or lose his footing and fall into the sinkhole, taking damage but avoiding the entanglement.
This is all infomation i have
The PC success a DC 13 Percepcion check and known the sinkhole
*If the hazard in question is not an impassable wall like structure and is triggered by someone entering one of the squares it occupies, then it is not triggered as the character technically did not enter a square occupied by the hazard. In the same way they wouldn't provoke an AoO from a creature that only threatens the squares occupied by the hazard.
This is the other GM tell me, but i argument in diagonal movement that step the 4 square.
Thanks
Based on the description it seems like the hazard wouldn't trigger from the diagonal movement shown. If instead of being a hazard, the box simply represented an area of difficult terrain, the character wouldn't take a penalty to their movement as a result of the diagonal movement shown.
Would the movement offer an attack of opportunity against the character to an enemy threatening the relevant square?
I believe it is the same rule that decides whether a character has passed via a square of not.
I don't like grids on battle maps because of this kind of things. I could leave every character and item at the same place and rotate the orientation of the grid with as a result a completely different play of the combat.