I can't seem to find anything that they did butt heads about.
Did they ever have any public disagreements that were recorded by history?
In 1806 James Monroe, at the time serving as the US minister to Britain, tried to renew the Jay Treaty of 1795. However, both Thomas Jefferson and James Madison (then Secretary of State) rejected the treaty. Monroe's and Madison's differing opinions about foreign affairs came up again in 1808, when Monroe was considered by parts of the Democratic-Republican Party as a presidential candidate, instead of Madison.
However, the two men's relations were amicable, to the point where Madison named Monroe his Secretary of State in 1811, and then Secretary of War in 1814.