<p>The Fibonacci number of a graph, introduced by Prodinger and Tichy, is defined as the total number of independent vertex sets, including the empty set, and it is closely related to the Merrifield–Simmons index in chemical graph theory. In this paper, we investigate the Fibonacci numbers of generalized Fibonacci graphs, a family studied in the context of source-to-sink directed acyclic graphs and related algorithmic questions. We show that the underlying undirected structure of these graphs coincides with a power of the path graph, which allows us to interpret independent sets as separated subsets of a linearly ordered vertex set. Using the standard vertex-deletion method, we derive a Fibonacci-type recurrence for the number of independent sets together with natural initial conditions. We further obtain an explicit closed form in terms of binomial coefficients and determine a rational ordinary generating function. In addition, we give an explicit expression for the independence polynomial, from which structural parameters such as the independence number and the number of maximum independent sets follow immediately. Several special cases are discussed, including the recovery of the classical Fibonacci numbers for paths, as well as degenerate complete-graph regimes. Finally, we derive Binet-type expansions, asymptotic growth estimates, and a Nordhaus–Gaddum-type relation involving the Fibonacci numbers of the graphs and their complements.</p>