In this paper, we study the Chern character operators on the equivariant cohomology of the Hilbert schemes of points in the complex affine plane $C^2$ with the action of the torus $(C^*)^2$, and partially verify Okounkov's Conjecture [Oko, Conjecture 2] in this setting. Our main idea is to apply the connection between the equivariant cohomology of these Hilbert schemes and the ring of symmetric functions, via the deformed vertex operators of Cheng and Wang [CW], (the integral form of) the Jack symmetric functions and the transformed Macdonald symmetric functions of Garsia and Haiman [GH, Hai].