Kion's grandfather did fall from a big branch. But he was saved by the Lion Guard of his day.
When Bunga steals Rafiki's staff, he stumbles upon paintings of Nala's father as a cub and mistakes him for Kion. Curious, Bunga touches the paintings with Rafiki's staff, bringing them to life, and watches in horror as Nala's father falls from a tree. A frightened Bunga proceeds to protect Kion from every tree they encounter, terrified that Rafiki's paintings predict the future.
Later, Rafiki reveals that the painting is in fact of Nala's father. In the past, Nala's father had fallen from a tree branch and been saved by the Lion Guard of his day.
Though only pictured as a cave drawing, Nala's father appears to be near-identical to his grandson, Kion. His fur is brownish-gold, with a dash of white on his chin, and his paws and underbelly are bright gold. Like his grandson, he has a bright red tuft of mane on his head.
The Lion King Wiki has a collection of images and media related to Nala's father.