In this paper, we prove that the global version of the Łojasiewicz gradient inequality holds for quadratic sphere constrained optimization problem with exponent ${\theta=\frac{3}{4}}$. An example from Ting Kei Pong shows that ${\theta=\frac{3}{4}}$ is tight. This is the first Łojasiewicz gradient inequality established for the sphere constrained optimization problem with a linear term.