Help with audio EQ cookbook BPF filters and Q
i believe that what you're seeing is the compression of bandwidth done by the frequency warping inherent with the bilinear transform.  now, if you look at cookbook, the bandwidth $BW$ is expressed in terms of octaves.  so, for your spec:

$$ BW = \log_2(20250) - \log_2(19750) = 0.036069 \text{ octave} $$

try using that number for $BW$ or the corresponding $Q$ as shown in the cookbook:

$$ \frac{1}{Q} = 2 \sinh\left( \frac{\ln(2)}{2} BW \frac{\omega_0}{\sin(\omega_0)} \right) $$

i come up with $Q = 7.63359$.

your formula relating bandwidth in linear frequency to Q is accurate for analog filters but is accurate for digital filters (designed using bilinear transform) only for frequencies much lower than Nyquist.  the reason why is that frequency warping of the BLT moves frequency points downward and even after compensating the center frequency, it compresses the bandwidth in comparison to the analog counterpart.  and 20 kHz is quite close to Nyquist, so we expect this frequency warping and bandwidth compression.

note that for analog, the equation relating $Q$ and $BW$ (in octaves) is slightly different:

$$ \frac{1}{Q} = 2 \sinh\left( \frac{\ln(2)}{2} BW \right) $$

note that the factor $\frac{\omega_0}{\sin(\omega_0)}$ increases the effect of bandwidth, so the digital bandwidth need be smaller than the analog bandwidth especially as $\omega_0 \to \pi = $ Nyquist.