  Quantum geometry plays a pivotal role in the second-order response of $\cal
PT$-symmetric antiferromagnets. Here we study the nonlinear response of 2D
altermagnets protected by $C_n\cal T$ symmetry and show that their leading
nonlinear response is third-order. Furthermore, we show that the contributions
from the quantum metric and Berry curvature enter separately: the longitudinal
response for all planar altermagnets \emph{only} has a contribution from the
quantum metric quadrupole (QMQ), while transverse responses in general have
contributions from both the Berry curvature quadrupole (BCQ) and QMQ. We show
that for the well-known example of $d$-wave altermagnets the Hall response is
dominated by the BCQ. Both longitudinal and transverse responses are strongly
dependent on the crystalline anisotropy. While altermagnets are strictly
defined in the limit of vanishing SOC, real altermagnets exhibit weak SOC,
which is essential to observe this response. Specifically, SOC gaps the
spin-group protected nodal line, generating a response peak that is sharpest
when SOC is weak. Two Dirac nodes also contribute a divergence to the nonlinear
response, whose scaling changes as a function of SOC. Finally, we apply our
results to thin films of the 3D altermagnet RuO$_2$. Our work uncovers distinct
features of altermagnets in nonlinear transport, providing experimental
signatures as well as a guide to disentangling the different components of
their quantum geometry.
