Altermagnets, as an emergent class of quantum magnets, have recently attracted considerable attention, yet an intrinsic transport hallmark of altermagnetic order remains elusive. Here, through spin-group and magnetic Laue-group symmetry analyses, we identify the third-order intrinsic anomalous Hall effect (IAHE) as such a hallmark and show that it is a spin-group-symmetry-breaking response, forbidden without spin-orbit coupling (SOC) but allowed for all ten altermagnetic spin Laue groups once SOC is included. Using the Lieb-lattice altermagnet and the experimentally realized ${\mathrm{V}}_2{\mathrm{Se}}_2\mathrm{O}$ as examples, we demonstrate resonant enhancement of the third-order IAHE near altermagnetic band crossings at generic momenta, leading to an experimentally observable Hall voltage of about $10\mu \mathrm{V}$. We further identify the Berry curvature quadrupole, activated by finite SOC, as the quantum-geometric origin of this enhancement. Our results establish the third-order IAHE as an intrinsic transport hallmark of altermagnets and complete the hierarchy of IAHE across collinear quantum magnets.

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• Anomalous Hall effect
• Berry curvature
• Quantum geometry
• Quantum metric
• Spin-orbit coupling
• Altermagnets
• Group theory
• Semiclassical methods