Wars are not fought between entire species, they are fought between various social groups.
And wars are fought for good or bad reasons, that are usually a lot stronger than merely "Some other group of people exists so let's attack them".
History has examples of cooperation between groups belonging to different species.
For example, there are instances where dolphin groups and human groups cooperate to hunt fish.
In a region of Australia, a group of orcas cooperated with aborigines to hunt whales, and later cooperated with European whalers. In this example, it is at least slightly possible that both the orcas and the whales they preyed upon had intelligence equal to that of the humans involved.
And I find it easy to imagine that there could have been examples of cooperation between groups of early humans and groups of proboscideans (relatives of elephants) to dig water holes, for example.
And at the present time there could occasionally be small scale wars between chimpanzee bands, or between bands of Chimps or gorillas and bands of humans.
And when humans hunt elephants and/or take over their habitat, elephants sometimes strike back, with single elephants or herds attacking human villages.
And when there were several species of genus Homo co-existing relations between various small bands of the same or of different species could have been just a complex and varied as relations between modern nations can be.
So I can imagine wars between two different tribes of Cro-Magnons, wars between two different tibes of Neanderthals, Wars between a tribe of Neanderthals and a tribe of Cro-Magnons, wars between an alliance of Cro-Magnon tribes and a Neanderthal tribe, wars between an alliance of Neanderthal tribess and a Cro-Magnon tribe, wars between an alliance of Cro-Magnon tribes and an alliance of Neanderthal tribes, and wars between an alliance of Cro-Magnon and Neanderthal tribes and another alliance of Cro-Magnon and Neanderthal tribes, as well as various periods of peace between various such groups.
So you, the writer, the creator god of your fictional universe, should ask yourself:
"Why should two completely different species, newly adjacent to each other, go to war?"
Because no war mongers who may want the government of part or all of their species to go to war against some government of part or all of the other species are going to convince enough high ranking leaders and ordinary people that war is a good idea, unless they have some sort of reason, good or bad, accurate or false, to make it at least seem to a lot of their people that the war will be a good idea, whether it actually is a good idea or not.
So if you, the creator god of your fictional universe, want war or peace to exist in your story, or for peace to be broken by war, or for war to be ended by a peace treaty, you can arrange for that to happen.
And if you want the readers to desire war, or to want peace, or to change their opinions about war and peace, you can make them feel the way you want them to, if you are a sufficiently skilled writer.
And if you want to have a war between the government and society of part of one species and the government and society of part of another species to happen, you should take the time to give the sides motives for going to war that seem strong enough, at the time, to them.