This work proposes a sub-equation approach to achieve new traveling wave outcomes for conformable Klein-Gordon equation. This approach is strong and efficient, and it only takes a few simple steps to solve. By using the appropriate transformations, the conformable Klein-Gordon equations transform into ordinary differential equations. The novel exact traveling wave solutions were obtained in the form of trigonometric and hyperbolic solutions. Furthermore, the results are drawn in two and three dimensions along with a contour plot, and the impact of fractional parameters on the waveform is examined with the help of Mathematica. Additionally, the dynamical assessment is explained in which bifurcation and sensitive analysis are discussed to understand the model’s dependability. It is demonstrated that the wave solutions’ dynamics offer a more meaningful perspective on the outcomes, helping the reader better understand the nonlinear wave equation that represents physical processes. Conformable Klein-Gordon equations have various applications in different fields such as solid-state physics, non-linear optics, quantum field theory, plasma physics, and mathematical physics. Examining solitary wave solutions can explore many physical relations that can be very helpful in different areas.