In this paper, we define generalized $\varphi-$quasi contraction map which is more general than strict quadratic quasi contraction map by using an altering distance function $\varphi$ and prove the existence and uniqueness of fixed points of these maps satisfying asymptotically regular property in the setting of complete metric spaces. We extend these results to $T$-orbitally complete metric spaces. Examples are provided to illustrate our results. Our results generalize Theorem 4 of [O. Popescu, G. Stan, \emph{Some fixed point theorems for quadratic quasi contractive mappings}, Symmetry, extbf{11} (2019)].