Anapole superconductivity recently proposed for multiband superconductors [S. Kanasugi and Y. Yanase, Commun. Phys. 5, 39 (2022)2399-365010.1038/s42005-022-00804-7] is a key feature of time-reversal () symmetry broken polar superconductors. The anapole moment was shown to arise from the asymmetric Bogoliubov spectrum, which induces finite center of mass momenta of Cooper pairs at the zero magnetic field. In this paper, we show an alternative mechanism of anapole superconductivity: the quantum geometry induces the anapole moment when the interband pairing and Berry connection are finite. Thus, the anapole superconductivity is a ubiquitous feature of -broken multiband polar superconductors. Applying the theory to a minimal model of , we demonstrate the quantum geometry induced anapole superconductivity. Furthermore, we show the Bogoliubov Fermi surfaces (BFS) in an anapole superconducting state and predict an unusual temperature dependence of BFS due to the quantum geometry. Experimental verification of these phenomena may clarify the superconducting state in and reveal the ubiquitous importance of quantum geometry in exotic superconductors.