Poland did not actually engage in any actions that helped to bring about the Cold War. Instead, Poland helped bring about the Cold War simply by existing where it did, to the west of the Soviet Union.
Early in WWII, Poland had been divided between Germany and the Soviet Union at the time that those two nations still had a nonaggression pact between them. Poles, not surprisingly, wanted their independence back. It was ostensibly to protect Polish independence that British and French went to war with Germany.
At the end of WWII, however, the Yalta Conference gave Poland to the Soviet Union. It had new borders that moved it farther west, giving it more of what had been Germany. The Soviets were supposed to allow Poland to hold free elections. They did not and instead imposed a communist government on Poland. The Soviets felt that they needed to control Poland as a buffer zone to protect them against invasions from the west.
To the US and Britain, the Soviets’ actions in Poland showed that they wanted to expand their power and to dominate everything around them. This made the Western Allies very suspicious of the communists. This suspicion helped to bring about the start of the Cold War.
Thus, Poland did play a role in the start of the Cold War, but it was a passive role.
After the Second World War, Poland was placed under the control of the Soviet Union until free elections could be held. A coalition government shared between, the Polish government-in-exile and the Communist Polish Workers Party was established. However, the elections that were held were skewed to ensure the Communist Party dominated the results. The Soviet Union was in favor of a Communist-led government, and they supported the destruction of the opposition. The Soviet Union’s activities in Poland were in clear contravention of the Yalta agreement, but the Western powers could not intervene.
The Soviet Union continued to control the Communist government, and Western powers sensed mischief in Soviet activities. On the other hand, the Soviet Union did not hide the fact that they wanted complete control of all territories around them and beyond. To achieve their plan, the Soviet Union wanted to install Communist governments in the different countries.
Thus, Poland served as a successful experiment for the Soviet Union with regards to the spread of Communism, which led to heightened suspicions, and the onset of the Cold War.