Would history be different if Britain never passed the quartering act? Why or why not?

Truth to be told, the Americans did not react immediately to the Quartering Act. Nor were they violent in response, at first.

In the end, The Quartering Act was received as "another example" of the Tyranny of Britain and of how they would use any excuse to get more money from the colonies. In the end, the USA would have fought for its independence anyways. If not then, later, but no country in the world remained a colony of the Europeans forever.

The Americans had plenty of reasons to plot a revolution and to fight for independence. This, The Quartering Act, was just one more reason.