The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.Fibonacci numbersFnand Lucas numbersLnbelong to a large integer family {xn}, often defined by the recurrencexn=xn−1+xn−2, wherex1=a,x2=b, andn≥ 3. Whena=b= 1,xn=Fn; and whena= 1 andb= 3,xn=Ln. Clearly,F0= 0 andL0= 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:In this article, we will give a brief introduction to theQ-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.In 1960 C.H. King studied theQ-matrix