For the first question - are you using nightly build of pytorch? The op name was updated in https://github.com/pytorch/pytorch/pull/36494 6 to quantized.batch_norm2d to make it more consistent with the implementation. You might have to re-do the QAT convert with the same pytorch build to make sure you get the same op name.
For the second question - We currently do not have the quantized pytorch to caffe2 conversion flow working for the quantized::batch_norm2d operator. Mainly due to the fact that caffe2 quantized ops currently don’t have this operator.