We investigated quantum Hall states in an inverted HgTe quantum well (QW) close to the critical thickness using transconductance fluctuation (TF) measurements. In the conduction band, several integer quantum Hall states were observed, corresponding to filling factors ν=1,2,3,4. For magnetic fields above 2 T, quantum Hall states ν=0 were observed in the normal gap. These observations agreed well with the previous studies of quantum Hall states on GaAs QWs and graphene. Interestingly, TFs corresponding to anomalous positive filling factor ν were clearly observed in the valence band. We attribute the emergence of those TFs to the localization and charging of the heavy holes located in the side maxima of the valence band.