Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC)  numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC  number theory can extend dual quaternions to HGC  dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC  Fibonacci/Lucas numbers, their special matrix correspondences have been identified and these are carried out to HGC  Fibonacci/Lucas dual quaternions. Furthermore, we provide a more accurate way to quickly calculate HGC  Fibonacci numbers and associate this with HGC generalized Fibonacci numbers. For implementation, we produce an algorithm in Maple. Lastly, we put the theory into practice.