We've had nearly continuous warfare for the last 2,000 years. I've read a number of history books on it, and none have suggested climate change as being a cause, or even a factor.
This article (written in backlash to some recent high profile claims of conflicts caused by climate change) suggests that blaming climate for human conflicts is nothing new:
It’s that they promote a kind of climate reductionism, one
that carries echoes of the deterministic theories that were
once popular during the second half of the 19th century.
Consider, for instance, John William Draper’s History of
the American Civil War, published in 1867. Draper, who
served as president of the medical college of New York
University from 1850 to 1873, was a professor of chemistry
and an architect of the so-called “conflict model” of
science and religion, bringing a scientist’s eye to his
task. On the very first page of his book, he announced “the
great truth that societies advance in a preordained and
inevitable course” guided by “uncontrollable causes.” Chief
among these was the climate.
You wouldn't ever see "the climate changed so we moved" as a cause here, because climate change takes a few generations to fully manifest. People without written records and good observation tools wouldn't be able to spot or record the climate change. But such people would certainly notice when agricultural output drops and decide to move to places where the situation is better.
Such wholesale migrations have been the cause of many conflicts. Steppe nomads have migrated to Europe on many occasions over the last two thousand years and each time it has led to war. Even within Europe this sort of migration was observed. For example, Julius Caesar fought Germanic tribes who had decided to migrate to Gaul because the grass was greener on that side of the Rhine.
Populations regularly outgrow their food supply and migrate. That isn't climate change. They also exhaust their land with poor farming practices, that isn't climate change, either.
A drought isn't climate change. The annual weather is a chaotic system, and a drought for a year or two (or even 10) is normal. Climate change is much longer term.