The Nature of the Beast Trophy in Hogwarts Legacy

1. The main quest The Elf, the Nab-Sack, and the Loom gives you the Nab-Sack for catching beasts at their dens, the paw prints on the map.
2. Deek's side quest Foal of the Dead has you buy a Breeding Pen spellcraft at Tomes and Scrolls in Hogsmeade and opens a Vivarium in the Room of Requirement. Conjure the pen there.
3. Catch a male and a female of each kind, put them in the Vivarium and breed them at the pen. A young beast takes 30 minutes.

The Unicorn has a single den, in the Forbidden Forest beside the Upper Hogsfield Floo Flame, and comes with the side quest The Unique Unicorn. Graphorns appear only after San Bakar's Trial, at a den on Clagmar Coast.