  We present results on the measured shapes of 832 galaxies in 11 galaxy
clusters at 1.0 < z <1.4 from the GOGREEN survey. We measure the axis ratio
($q$), the ratio of the minor to the major axis, of the cluster galaxies from
near-infrared Hubble Space Telescope imaging using S\'ersic profile fitting and
compare them with a field sample. We find that the median $q$ of both
star-forming and quiescent galaxies in clusters increases with stellar mass,
similar to the field. Comparing the axis ratio distributions between clusters
and the field in four mass bins, the distributions for star-forming galaxies in
clusters are consistent with those in the field. Conversely, the distributions
for quiescent galaxies in the two environments are distinct, most remarkably in
$10.1\leq\log(M/{\rm M}_{\odot})<10.5$ where clusters show a flatter
distribution, with an excess at low $q$. Modelling the distribution with oblate
and triaxial components, we find that the cluster and field sample difference
is consistent with an excess of flattened oblate quiescent galaxies in
clusters. The oblate population contribution drops at high masses, resulting in
a narrower $q$ distribution in the massive population than at lower masses.
Using a simple accretion model, we show that the observed $q$ distributions and
quenched fractions are consistent with a scenario where no morphological
transformation occurs for the environmentally quenched population in the two
intermediate mass bins. Our results suggest that environmental quenching
mechanism(s) likely produce a population that has a different morphological mix
than those resulting from the dominant quenching mechanism in the field.
