The quantization of neural networks for the mitigation of the nonlinear and components' distortions in dual-polarization optical fiber transmission is studied. Two low-complexity neural network equalizers are applied in three 16-QAM 34.4 GBaud transmission experiments with different representative fibers. A number of post-training quantization and quantization-aware training algorithms are compared for casting the weights and activations of the neural network in few bits, combined with the uniform, additive power-of-two, and companding quantization. For quantization in the large bit-width regime of $\geq 5$ bits, the quantization-aware training with the straight-through estimation incurs a Q-factor penalty of less than 0.5 dB compared to the unquantized neural network. For quantization in the low bit-width regime, an algorithm dubbed companding successive alpha-blending quantization is suggested. This method compensates for the quantization error aggressively by successive grouping and retraining of the parameters, as well as an incremental transition from the floating-point representations to the quantized values within each group. The activations can be quantized at 8 bits and the weights on average at 1.75 bits, with a penalty of $\leq 0.5$~dB. If the activations are quantized at 6 bits, the weights can be quantized at 3.75 bits with minimal penalty. The computational complexity and required storage of the neural networks are drastically reduced, typically by over 90\%. The results indicate that low-complexity neural networks can mitigate nonlinearities in optical fiber transmission.