Sadly not. The set of languages typically available to PCs is listed in Ch. 4 of the Players Handbook. Players typically select from either the Standard or Esoteric languages.
Standard Languages include Common, Dwarvish, Elvish, Giant, Gnomish, Goblin, Halfling, Orc as well as any regional language like Chultan.
Esoteric Languages include Abyssal, Celestial, Draconic, Deep Speech, Infernal, Primordial, Sylvan and Undercommon. These languages are typically learned by scholars, and are used by creatures from other planes for the most part.
Two languages known by specific classes would be Thieves Cant for rogues, which is more of a patois than a language, and Druidic for druids, of course.
Beast Speech is a feature of Forest Gnomes, which allows them to communicate with small animals, which notably do not have a language in their stat blocks.