Related: When did empire and imperialism become dirty words?

I think that it depends on which part of the globe you want to focus. In Europe, I would suggest the last great campaigns of "conquest" ended with the exile of Napoleon. A dominant faction of French revolutionaries sought to extend the revolution beyond the borders of France, a natural revolution across Europe. French territory no longer necessarily ended at the "natural frontier" of the Rhine. Previously the French monarchy and Austria or Spain would squabble over Alsace-Lorraine or the Low Countries, but these could be considered minor territorial disputes that happened to involve multi-year campaigns as opposed to massive wars of conquest.

I would suggest that the revolutionary decade (1789-1799) allowed a shift in the attitudes of Frenchmen that Napoleon was able to harness. While some of his campaigns (Wars of the third and fourth coalitions) were aimed at knocking out great powers (Austria and Prussia successively), many of his campaigns were about conquest, namely Italy, Egypt and Spain. While the great powers were neutered and allowed to remain as figurehead allies, he crowned his own Kings in Spain (his brother Joseph) Naples and Sicily (Joseph again, as well as his Marshall Murat), etc.

The age of Napoleon created two major forces. The first was the shift from smaller field armies which fought tidy battles/wars to total mobilization of a nation's forces and a war footing economy. In Napoleon's victory at Austerlitz, which many consider his most impressive, he commanded 65,000 men against about 85,000. At Leipzig, "The Battle of Nations", in 1813, he commanded about 200,000 men, against about 350,000 men. The physical size of the battlefield grew exponentially, as did the number of cannon involved. The point here is that the nature of warfare changed. It became less "orchestrated", and governing a conquered territory became a much more serious proposition. Napoleon found this out in Spain, with the Peninsular wars giving rise to the term "guerilla warfare". National pride came into play. Total war was practiced on a larger scale.

The second major force came after his exile. The Concert of Europe, or Age of Metternich, saw a balance of great powers and general peace and stability in Europe for a few decades. One cannot understate how devastating the Napoleonic wars were to Europe, but even as the sixth coalition was pushing Napoleon back to France, Klemens von Metternich, Austrian diplomat, was fearful of a mobilized Russia, and possibly sought to keep Napoleon in power, simply to keep the continental forces in relative equilibrium. Metternich, Tallyrand, Lord Castlereagh, et al kept the peace through diplomacy rather than war.

It could be argued that the Age of Metternich lasted to the Crimean War in 1853. This would be more than a generation of general peace on the continent, which is almost unthinkable. A point I would add here but am getting out of my depth is that by 1848, the internal dynamics within the remaining great powers made international conflict a dicey proposition. 1848 was the year Marx and Engels put out the Communist manifesto...urbanization in France and other places put an emphasis on worker rights, economic inequality, etc. These forces prevented kings from playing at war as in previous centuries.

I would target this era, from 1815 through 1848, as the general death of "conquering" in Europe.

Now think about the happenings in America at the same time. Manifest Destiny. The United States was expanding westward. The Mexican-American War, which I would argue was a naked war of conquest, took place from 1846 to 1848. Polk campaigned in 1844 on the slogan of "54 40 or fight" which was in reference to US claims of Oregon Country. The US nearly went to war with Britain to claim territory in what is now British Columbia. In this sense, conquering was very much in fashion in the US during the same time that it was falling out of fashion in Europe.