Storm means attack.
Now, both conquer the Justice and and storm the Justice would sound wrong to us in most contexts, since the main meanings of justice are the abstract noun meaning rightness based on ethics or law, and the concrete noun meaning a judicial officer (the precise meaning differing in different countries).
If we'd already established that "The Justice" was being used as a proper noun as the name of a particular fortress, as you say, then that would give us a context where both conquer the Justice and storm the Justice would be grammatical.
The meaning would be the same though. Storm would apply to an attack, whether successful or not, or indeed if it was a lightening attack intended to do damage or obtain some other strategic goal, without taking the fortress.
Conquer would only apply to attacking, taking, and then continuing to occupy the fortress, and defending it against any counter-attack or insurrection from the defeated populace.
The most common verb to refer to attacking a fortress (or a castle, walled city, etc.) and gaining control of it, is take. Hence Take the Justice.
It still sounds strange though. Again, this could be fine if the context establishes "the Justice" as a name of a particular place, or you might deliberately let it sound strange, with the meaning becoming clearer later on.