The mixing condition is often considered to modeling the functional time series data. Alternatively, in this work we consider the problem of nonparametric estimation of the regression function in Single Functional Index Model (SFIM) under the quasia-ssociation dependence condition. The main result of this work is the establishment of the asymptotic properties of the estimator, such as the almost complete convergence rates. Furthermore, the asymptotic normality of the constructed are obtained under some mild conditions. We finally discuss how to apply our result to construct the confidence intervals. Finally, the finite-sample performances of the model and the estimation method are illustrated using the analysis of simulated data.