Strangely enough, there sort of was, but it wasn't exactly a movement. It more or less stemmed from a single man's ambitions.

The nineteenth century saw the world divided into large racial blocs, at least in the eyes of European intellectuals. In a lot of places, the idea of a "Latin" race became very popular. The French, Italians, Spanish, Portuguese, and some others were all seen as bound together through speaking the languages descended from the Latin of the Romans, as well as united under a single, Catholic faith. According to these intellectuals, like Charles Maurras, the several racial blocs in Europe and the Americas had very distinct differences that pitted them against one another. There were the Latin peoples, the Anglo-Saxon peoples, the Germanic peoples, and the Slavic peoples.

While many of these intellectual developments mainly came in the late nineteenth century, particularly in the wake of the Dreyfus Affair, their roots go further back to around the 1830s. But one of the idea's early proponents was the French emperor Napoleon III. While his uncle Napoleon had been briefly successful at bringing much of Europe under his control, Napoleon III's ambitions were much larger than that. It was one of the emperor's goals to create and sustain a new French empire that would be state by, for, and of the Latin race.

And it was this goal that inspired Napoleon III to maintain ties with French-speaking regions of the Americas, like Quebec and Louisiana. Furthermore, the French intelligentsia and political establishment began to group all speakers of Latin-based languages together in a geopolitical sense. And hence we have the creation of "Latin Europe". But, more importantly, they did this for speakers of French, Spanish, and Portuguese in the Americas. Before this time, the countries south of the United States was simply called Hispanic America. But with Napoleon III's imperial ambitions, we see for the first time the creation of Amérique Latine, or Latin America. The invention of Latin America by French imperialists was the justification behind France's intervention in Mexico in the 1860s (which is why Cinco de Mayo is celebrated). The sense of Latin Europe and Latin America was promoted to facilitate French imperialist and hegemonic ambitions. Napoleon III's pan-Latinism may have also been the reason behind his support for Italian unification.

Even though our conception of Latin America is very different from the French pan-Latinists' conception, their use of the term and the actions it justified dramatically altered the history of the region. If you want to read more about Pan-Latinism in France, John Leddy Phelan wrote a wonderful article outlining the history of the movement particularly in relation to France's intervention in Mexico.