Spectral Property of Bandlimited Quantized Signal
I guess this probably just a mistake in your analysis code.
Quantization noise is white and for a noise signal it's uncorrelated to the original signal, so spectrum of the original noise doesn't matter.
I did repeat your steps and saw exactly what I expected: The quantization noise is white and the spectrum of the quantized signal follows the original signal at low frequencies and is white at high frequencies (since it's dominate by the quantization noise).
Potential errors in your code

Clipping the signal
Improper spectral estimation, i.e. wrong  choice of window and/or overlap. Not windowing or properly during spectral analysis. Either window manually or just use PWELCH instead.

Here is a MATLAB code snippet that shows this
%% script to generate quantization noise
nFFT = 16384;
fs = 48000; % sample rate
fc = 4000;  % cut off for low pass filter

% create noise, filter, and normalize
x0 = randn(10*nFFT, 1); % 10 frames or so
[b,a] = butter(4,fc*2/fs);
x0 = filter(b,a,x0);
x0 = x0./max(abs(x0));

% quantize to 8 bit
xq = round(2.^7*x0)*2^(-7);

% calculate quantization noise
noiseSignal = xq-x0;
% power spectrum
yAll = [x0 xq noiseSignal];
psd = pwelch(yAll,hamming(nx));
freqAxis = fs*(0:nFFT/2)'/nx;
% graph
clf; figure;
plot(freqAxis,10*log10(psd));
title('Power Spectra');
ylabel('Level in dB');
xlabel('Frequency in Hz');
legend('Original','Quantized','Noise');
grid ('on');