the Borg speak Latin

The first intuitive answer would be that the Borg got Latin from somebody who speaks Latin. But people who speak Latin are slightly more rare in Star Trek than they are in the real world.

However, any significant Federation computer should contain all the vocabulary and syntax of all known languages (the universal translator works much more efficiently when it knows what it's talking about). Throughout their conquests in the Alpha Quadrant they would have assimilated the data banks of at least one computer that knew how to speak Latin.

There are indeed very few people who speak Latin in Star Trek. But Jean-Luc Picard happens to be one of them.

When he got assimilated in 2366 and his mind became a part of the collective consciousness, the Borg gained his knowledge of this ancient language (oh, and some tactical stuff as well).

Locutus of Borg became a voice for the Borg towards the Federation, as the Latin name implies.