I apologize if some of this seems like stuff I've said recently, but one of my games recently entered an interesting political phase so - shockingly - this is on my mind. It also doesn't help my new found love of Sigma Theory and Diplomacy are keeping it in mind.
That said, let's talk about why countries and kingdoms would go to war in your game.
Peace must be maintained, ergo it is not the status quo...
It is more or less in the title, but on the international stage - or even interstate depending how big a country is and how much infighting is allowed - peace is a thing that has to be maintained. And since peace has to be maintained, it can't be the natural state. After all, left alone things will devolve into violence and war.
That said, in the real world there are three reasons countries go to war: Honor, Fear, and Pride.
Luckily, for you in fiction you have a fourth reason: some madman or woman is pushing for war for some reason or another. For example, in For Honor, Apollyon is keeping the three kingdoms at war thinking that the constant struggle of war will help make humanity as a whole stronger. Now, I said this was a fourth way, but really this is just someone manipulating the above three reasons.
Honor is the weirdest to me. Mostly because it sounds like an agreement to go to war. And then I realized that is exactly what a treaty is, and that is also why some treaties are not full treaties but defensive pacts - meaning they only come into play in the event a kingdom is attacked from an outside source.
A country or kingdom goes to war anytime it needs to to honor its agreements. Alternatively this can be considered the reason if it is in response to an insult - though that can also be covered by fear.
Fear can also seem weird. Afterall, what does a kingdom have to fear? And then I played Diplomacy and I realized just what there is to fear. Every other kingdom - even allies - can be something to fear. Any neighboring country getting stronger can be a threat to a kingdom.
Neighbor getting bigger? How long before they need to grow into your lands? Neighbor building up an army? How long until they use that army to take what you have that they need? You can trust your allies, right? Sure, but are you really going to trust a person amassing troops near your borders that they're not going to move into your lands? And even if their leadership isn't telling them to...armies need to forage, and that is risking taking resources from your land. Not to mention awol soldiers turning to banditry or if there is a fight people fleeing into your lands.
That doesn't even begin to talk about the more existential fears like losing your position in the political, military, or other hierarchies with other nations. Why did Sparta go to war with Athens? Well, lots of reasons, but one of them was fear of losing their position of power as extended peace helped make Athens an economic powerhouse while they were more the old way military status.
Fear is a big motivator. Especially when combined with...
Nationalism, Patriotism, and other isms can be very dangerous to a country's neighbors. After all, what good is having the best army in the land if you never deploy it? How do you prove you are the better force, the better people, or the better whatever if you don't go out and actually prove it? And if you don't prove it, why would people believe it?
Most things are actually a combination of the three. For example, an insult to the nation is honor because that is what you are likely insulting - the honor of the kingdom. However, it is also pride, because that is what demands a response to an insult. Finally, it is also fear because letting an insult lie could cause a loss of status, and a loss of status is something to be feared.
Manipulating these reasons is how you cause conflict in your game that leads to war. The reason also gives a way for the PCs to be peace keepers, or peace makers.
Consider the following:
Kingdom A has been dealing with bandits. It is discovered the 'bandits' are actually people from a neighboring country. What does Kingdom A do? They have to respond. What happens when one of the raiders is the prince of Kingdom B? or a Count or Baron or some other noble? What happens when that important person is killed on foreign soil?
Or the situation I have in one of my games. The PCs are sure that the capital of Kingdom A is going to be attacked by the enemies they've been chasing. One of the PCs is the monarch of Kingdom B. He's bound by blood oath to not move his troops into Kingdom A without an invite. Now if Kingdom A falls, the enemy will come to B. So he needs to amass defenses just in case. But in doing so, Kingdom A just sees armies amassing at a border. Can they trust nothing is going to happen? If they respond that could weaken their ability to fend off the coming assault. So what do they do? And if things go wrong, who takes the blame?
Finally, consider your PCs claim an area for their own. Being PCs they invest into this area. They get market deals. They arrange with the thieves guild for smuggling. They build up their area. They pour their money from adventuring into it. You suddenly have this new place booming with money, drawing attention, and becoming a real presence. Worse, or better, the people running this place are able to do things - like taking down adult dragons that have fended off entire armies. If left to grow this place will become the real power in the area, destabilizing everything else, and knocking someone off their place at the top of the food chain...unless, said place takes out the PCs location first. Obviously a 'real' reason would need to be fabricated, but that fear alone can be impetus to find one.