In this manuscript, fractional order is introduced onto a time-delay differential equation model of cell-to-cell spread of HIV-1. The fractional derivative of Caputo type is considered. We deal with the local stability of the resulting system and derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this system. Explicit expressions for determining stability of critical surfaces are also given. An Adams-type predictor-corrector technique is applied to illustrate the numerical results. The main target of this study is to describe the structure of HIV-1 by using a fractional-order mathematical model, and the motivation of using fractional derivatives is the ability of these operators to capture memory effects in the system.