Some characters get to trigger an effect if they move in a particular way.

Can a character Move 0 squares and still trigger effects like this?

Two simple examples:
Volstagg
WARRIORS THREE, VOLSTAGG LEADING THE CHARGE: Charge, Plasticity. When Volstagg uses Charge and hits an opposing character, after resolutions each friendly character with the Asgardian keyword adjacent to him at the start
or
Tiger Shark
BLOOD IN THE WATER: Tiger Shark can use Charge. If Tiger Shark begins or ends a move action occupying water terrain, he can be given a close combat action as a free action after actions resolve.

Yup. Charging zero spaces is a valid move BUT! If you are trying to Charge a character that is already adjacent you MUST breakaway first. (Barring powers or abilities that allow you to move without breaking away, of course)

thanks... I definitely did not know that. it seem counter intuitive at first glance... but there you go

But, what about another figure?
Like, for example, the Hulk of Mighty Thor Fast Forces.

HULK SMASH: When Hulk moves, after actions resolve he can use Quake at no cost.

If he try to break away for move, and he doesn't make it, he can still make the free quake? He doesn't move at all.

Thank you for the answer.

No, Hulk won't be able to use Quake. The reason being, as you said, Hulk failed his breakaway roll and didn't move, so Quake won't trigger.

Even if somehow the game did not allow "moving zero squares", you can almost always move at least 2 squares. Which means you can move one square away, and one square back. So yeah, you can move zero squares, altho you need to breakaway first if you're adjacent to someone.