As the word suggests, a quarter-sphere is a quarter of a sphere, and thus the volume of a quarter-sphere is a quarter of the volume of the sphere.
The volume of a sphere is computed as
[tex] \dfrac{4}{3} \pi r^3 [/tex]
So, the volume of a quarter-sphere is
[tex] \dfrac{1}{4}\cdot \dfrac{4}{3} \pi r^3 = \dfrac{1}{3} \pi r^3 [/tex]
