There are different historical interpretations to when the Cold War actually began - was it in 1918, when the West fought against the Bolsheviks in the Russian Civil War? Was it at Potsdam in 1945, where disagreements between superpowers came about after the Second World War and no military alliance against Hitler was needed? Or was it in 1946 when Kennan introduced containment and Churchill declared the existence of an Iron Curtain between Eastern and Western Europe, within a month of each other? The origins of the Cold War can be argued for any of the above views, and thus is not definite; however, it is evident that ideology was an insignificant factor in all the above cases.
The Russian Civil War, where West faced off the Reds in direct military conflict, is an example of conflict due to ideology. Western beliefs in capitalism, democracy, and a free market economy were the complete opposite to communist ideology, which was about a one-party state where all was state-owned.
A communist Russia would be a threat to the interests of the West. Although ideology was an important factor here, to say that this sole event was the origin of the Cold War would be ignoring significant events that occurred in following years.
In 1933, Roosevelt became president of the US and recognized the USSR. If ideology were so important, would the most prosperous capitalist country in the world acknowledge a communist Russia? When the US entered WWII in 1941, the two superpowers were allies - ideology was not a concern here; victory against fascism was.
Though it is true many problems resulted from the ambiguous agreements at Yalta, and along with change in leadership of two of the Big Three led to many disagreements at Potsdam, these were not related to ideology.