Electrical current fluctuations provide a powerful and unconventional probe of band geometry in quantum materials. In particular, intrinsic noise components that are independent of the relaxation time reveal universal band-geometric properties of Bloch electrons. Here, we identify a distinct intrinsic contribution to current noise at second order in the electric field, which is governed by the quantum metric. This effect arises from field-induced modifications to Bloch wavefunctions and band energies, and it dominates in PT-symmetric antiferromagnets where Berry curvature based contributions are forbidden by symmetry. Applying our theory to CuMnAs, we demonstrate that thermal noise provides a direct signature of quantum metric in PT-symmetric antiferromagnets.