Developing a model of fractional differential systems and studying the existence and stability of a solution is considered one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to presenting a hybrid system of delay fractional differential equations with boundary integral conditions, namely Atangana-Baleanu-Caputo (ABC) differential equations. Also, the fixed point (FP) technique has been applied to investigate the existence of solutions for the proposed hybrid system. Moreover, stability results have been studied for the solution of the desired system in the sense of Hyers-Ulam (HU). Finally, to support our findings, we provide two examples with different parameter values.