We introduce modified Noor iterative method in a convex metric space and apply it to approximate fixed points of quasi-contractive operators introduced by Berinde . Our results generalize and improve upon, among others, the corresponding results of Berinde , Bosede  and Phuengrattana and Suantai . We also compare the rate of convergence of proposed iterative method to the iterative methods due to Noor , Ishikawa  and Mann . It has been observed that the proposed method is faster than the other three methods. Incidently the results obtained herein provide analogues of the corresponding results of normed spaces and holds in $CAT(0)$ spaces, simultaneously.