If I want to say ‘the government is up to no good’, should I use a capital ‘g’ for government?

Not according to the new Australian Government Style Manual. 

But there’s a capital ‘G’ in the previous sentence, so when does government take a capital? 

The rule is that you use an initial capital for the word ‘government’ if it is part of a formal name, and use lower case everywhere else. 

It’s the same as the general rule for capitals. Use capitals for proper nouns and lower case for common nouns. 

Nouns are people, places, things, ideas, actions or concepts. Proper nouns are the names of specific people, places, things, organisations and so forth. Common nouns are all the rest.

The term ‘Australian Government’ is a proper noun because it is a name. So it has capitals. But when I write ‘all members of the government are rogues’, I am not using ‘government’ as a specific name, but as a general category.

The rule is simple, but many people break it. For some reason many people like to capitalise words that seem important, even if they’re common nouns. Government is one of the words most often offended against. (Of course this doesn’t apply at the start of a sentence, where you always use a capital, whatever the word.)