I don’t know the situation for you, but in general, in the US, the answer is YES. Most commonly this takes the form of lawsuits between federal and state/local governments where different levels of govt controlled by different people are looking to use the judiciary to enforce some rule or regulation they think the other is ignoring.
However (and i regret that I cannot recall a specific) I’m almost certain that a part of the same govt COULD sue another part. It obviously wouldn’t happen under normal circumstances, because all parts of the govt are under the direction of the same executive politician at any given moment in time. And so any disagreements would be resolved by appealing up the command chain.
But there are some ‘independent” parts of the government that could be external to that. In theory, this could lead to one part of the government suing another. But even that would be rare because those “independent” parts usually have authority over their little fiefdoms with sparse overlaps to another agency’s domain.
But there’s no reason it COULDN’T happen. It’s just really really unlikely that it WOULD.