  The current study examines the geometry of static wormholes with anisotropic
matter distribution in context of modified $f(\mathcal{G})$ gravity. We
consider the well known Noether and conformal symmetries, which help in
investigating wormholes in $f(\mathcal{G})$ gravity. For this purpose, we
develop symmetry generators associated with conserved quantities by taking into
consideration the $f(\mathcal{G})$ gravity model. Moreover, we use the
conservation relationship gained from the classical Noether method and
conformal Killing symmetries to develop the metric potential. These symmetries
provide a strong mathematical background to investigate wormhole solutions by
incorporating some suitable initial conditions. The obtained conserved quantity
performs a significant role in defining the essential physical characteristics
of the shape-function and energy conditions. Further, we also describe the
stability of obtained wormholes solutions by employing the equilibrium
condition in modified $f(\mathcal{G})$ gravity. It is observed from graphical
representation of obtained wormhole solutions that Noether and conformal
Killing symmetries provide the results with physically accepted patterns.
