Nambu-Goldstone bosons arise from the spontaneous breaking of continuous global symmetries, not from quantum anomaly symmetry breaking (which typically leads to explicit breaking or anomalous violations of conservation laws).
The retrieved literature standardly defines Nambu-Goldstone bosons as a consequence of spontaneous symmetry breaking (Goldstone's theorem), whereas quantum anomalies explicitly break symmetries at the quantum level, preventing them from being conserved and generally not yielding standard Goldstone bosons in the same manner. Therefore, the claim is refuted.