Is it ever appropriate for a society to question its government? If so, when?

Yes, it is appropriate for society to question its government. In fact, we should go further. Not only can citizens question their government, they should, and they must.
The most obvious time when citizens should question their government is when they think the government is doing something that is both ethically wrong and important. Staging a major protest against a minor government action (say, setting a speed limit fivr miles too high) is probably a waste of time and ethical authority.
On the other hand, when your government does something major that is very wrong, citizens must question and challenge their government. The easiest example of this from American history is the abolitionist movement. Slavery was legal. It was protected by law and tradition. Those who sympathized with slaves protested slavery, and those who were slaves, or had been slaves, challenged the government. Those who thought slavery was ethically wrong challenged the US government and ultimately succeeded in abolishing slavery.
https://www.history.com/topics/black-history/abolitionist-movement