A quadruped was a four-legged animal.
Doctor Phlox professed to Jonathan Archer that he had underestimated Archer's attachment to his "subservient quadruped". (ENT: "A Night in Sickbay") Later, when the command area was temporarily placed at the catwalk, Trip Tucker informed the captain that "the entire crew's accounted for including one quadruped" – his Beagle, Porthos. (ENT: "The Catwalk")
In his "Ode to Spot" in 2369, Data noted that Spot was an "endothermic quadruped, carnivorous by nature." (TNG: "Schisms", "A Fistful of Datas")
The Lan'Tuana sector was inhabited by quadrupeds. (VOY: "Bliss")
According to a portion of a deleted scene of the Star Trek: Deep Space Nine episode, "For the Uniform", Michael Eddington had once told a great story about a four-legged Talorian and a Orion slave girl to Julian Bashir and Miles O'Brien. (Star Trek: Deep Space Nine Companion - A Series Guide and Script Library)