  The interaction between dark matter and dark energy has become a focal point
in contemporary cosmological research, particularly in addressing current
cosmological tensions. This study explores the cubic Galileon model's
interaction with dark matter, where the interaction potential in the dark
sector is proportional to the dark energy density of the Galileon field. By
employing dimensionless variables, we transform the field equations into an
autonomous dynamical system. We calculate the critical points of the
corresponding autonomous systems and demonstrate the existence of a stable de
Sitter epoch. Our investigation proceeds in two phases. First, we conduct a
detailed analysis of the exact interacting cubic Galileon (ICG) model, derived
from the precise solution of the equations of motion. Second, we explore an
approximate tracker solution, labeled TICG, assuming a small coupling parameter
between dark matter and dark energy. We evaluate the evolution of these models
using data from two experiments, aiming to resolve the tensions surrounding
$H_0$ and $S_8$. The analysis of the TICG model indicates a preference for a
phantom regime and provides a negative coupling parameter in the dark sector at
a $68\%$ confidence level. This model also shows that the current tensions
regarding $H_0$ and $S_8$ are alleviated. Conversely, the ICG model, despite
its preference for the phantom regime, is plagued by an excess in today's
matter density and a higher expansion rate, easing only the $H_0$ tension.
