The generalized (3+1)-dimensional fractional q-deformed tanh-Gordon model and its optical solutions are studied under various physical conditions. Verifying our analytical findings and using the modified Sardar-sub equation technique (SSET) to examine the behavior of the governing model through convergence criteria under different parameters of the hyperbolic local derivative. By generating diverse optical phenomena and flexibility of the governing model, our findings pave the way for additional theoretical and applied physics research. The objective is also to study the bifurcation and chaotic behavior of the equation. To accomplish this, a dynamical system is created using the Galilean transformation. Analyzing the bifurcation and chaotic structure of the equation to identify important transitions that lead to chaotic behavior, and using phase-space analysis to understand the system's unpredictability. The study shows how small adjustments can have a significant effect on the results. This study clarifies the behavior of the model, which is crucial for several applications in quantum mechanics, physics, and optics.