ANAIS is a direct dark matter detection experiment whose goal is to confirm or refute in a model independent way the positive annual modulation signal claimed by DAMA/LIBRA. Consisting of 112.5 kg of NaI(Tl) scintillators, ANAIS-112 is taking data at the Canfranc Underground Laboratory in Spain since August, 2017. Results corresponding to the analysis of three years of data are compatible with the absence of modulation and incompatible with DAMA/LIBRA. However, testing this signal relies on the knowledge of the scintillation quenching factors (QF), which measure the relative efficiency for the conversion into light of the nuclear recoil energy with respect to the same energy deposited by electrons. Previous measurements of the QF in NaI(Tl) show a large dispersion. Consequently, in order to better understand the response of the ANAIS-112 detectors to nuclear recoils, a specific neutron calibration program has been developed. This program combines two different approaches: on the one hand, QF measurements were carried out in a monoenergetic neutron beam; on the other hand, the study presented here aims at the evaluation of the QF by exposing directly the ANAIS-112 crystals to neutrons from low activity $^{252}$Cf sources, placed outside the lead shielding. Comparison between these onsite neutron measurements and detailed GEANT4 simulations will be presented, confirming that this approach allows testing different QF models.