  Quintessence models have been widely examined in the context of
scalar-Gauss-Bonnet gravity, a subclass of Horndeski's theory, and were
proposed as viable candidates for Dark Energy. However, the relatively recent
observational constraints on the speed of gravitational waves $c_{\textrm{GW}}$
have resulted in many of those models being ruled out because they predict
$c_{\textrm{GW}} \neq c$ generally. While these were formulated in the metric
formalism of gravity, we put forward a new quintessence model with the
scalar-Gauss-Bonnet action but in Weyl geometry, where the connection is not
metric compatible. We find the fixed points of the dynamical system under some
assumptions and determine their stability via linear analysis. The past
evolution of the Universe can be reproduced correctly, but the late Universe
constraints on $c_{\textrm{GW}}$ are grossly violated. Moreover, at these later
stages tensor modes suffer from the gradient instabilities. We also consider
the implications of imposing an additional constraint $c_{\textrm{GW}} = c$,
but this does not lead to evolution that is consistent with cosmological
observations
