What is QEEG technically, and how is it computed?
qEEG or quantitative EEG is an umbrella term that encompasses many different signal processing methods that compute a number/statistic/quantitative biomarker from EEG signal traces. Algorithms usually boil down to some form of spectral analysis either using Fourier transforms, or wavelets.

The images in shown in the question are most likely generated through spectral analysis of individual EEG channels. In your figure, it seems like the author chose Delta to mean 1-3.5Hz. Some authors use Delta as all frequencies &lt;4Hz. (https://en.wikipedia.org/wiki/Electroencephalography)

To compute the contribution of each band, one resorts to standard power spectrum analysis techniques. For example, for a given EEG channel $i$, the Delta qEEG biomarker value can be computed as:

$$
\mbox{Delta}_i = \frac{\int_{1}^{3.5} P_{i}(f) df}{\int_{0}^{\infty}P_i(f)}
$$

where $P_i(f)$ is the power spectral density of the $i^{th}$ EEG channel signal. In practice, you can use, say, Welch method to estimate $P_i(f)$ from sampled EEG channel traces, the integrals will get replaced by sum over frequency bins, and you'd only go up to Nyquist.

Repeat this process for each frequency band, for each EEG channel to get sequences $\{\mbox{Delta}_i\}_{i=1}^N$,$\{\mbox{Theta}_i\}_{i=1}^N$, $\{\mbox{Alpha}_i\}_{i=1}^N, \cdots$, where $N$ is the number of EEG channels. 

The final step is to map these numbers on the image of the scalp. This is a visualization step which involves interpolation. You can use a tool like this: https://martinos.org/mne/stable/generated/mne.viz.plot_topomap.html#mne.viz.plot_topomap or https://www.nbtwiki.net/doku.php?id=tutorial:plot_a_time-frequency_representation_and_power_spectrum_of_a_single_channel#.WJtLMxC2pJI.