According to your links, this seems to be a CW radar that sends out pulses of a carrier frequency and demixes the received signal in a simple homodyne architecture, although as mentioned it would definitely be much easier and probably necessary to get more information from the manufacturer.
First, you have to possibly amplify(?), filter and sample the I and Q-path signals correctly and "isolate", i.e., cut out the samples from a transmit pulse based on the timing provided. Then you combine the paths as in $s[n] = I[n] + \mathrm{j}Q[n]$. I denote $\mathrm{j} = \sqrt{-1}$ as the imaginary unit. You combine I and Q-paths this way to be able to receive a complex-valued baseband signal.
CW radar operates directly based on the Doppler shift of moving objects so that if there is movement from a single source the signal of a single sampled pulse would be something like
\begin{equation}\label{eq:sig}\tag{1}
s[n] = \exp(\mathrm{j}2\pi f_D \mathrm{T_s} n) + \nu[n] \,,
\end{equation}
where $\nu[n]$ is assumed AWGN, $\mathrm{T_s}$ is the sampling time and the Doppler shift frequency $f_D$ is
\begin{equation}
f_D = \dfrac{\Delta v}{c} f_0
\end{equation}
with $c$ being propagation velocity (basically the speed of light), $f_0$ the carrier frequency of the transmit signal, and $\Delta v$ the relative radial velocity of the object.
Ideally all you need to do at this point is to detect or estimate this frequency, e.g. with FFT and CFAR-based processing as you mentioned. If there is some strong enough frequency peak(s) present, there is (radial) movement. You might want to process several pulses sequentially to be more reliable in your detection of movement.
Note that you could technically just sample and process the real-valued signal from a single path and could still detect movement with much reduced complexity. The disadvantage would be a lack of being able to distinguish direction of movement (i.e., sign of frequency) and a 3dB loss in useful signal power as compared to noise. For the full IQ processing you need to have two ideally identical paths of (amplification-)filtering-sampling and then combine the sampled data as described. If you only take the I-path and sample that single output, you just get the real value of the complex baseband signal, i.e., $\cos(\cdot)$ instead of $\exp(\mathrm{j}(\cdot))$ in \eqref{eq:sig}. You can of course still detect frequencies in this signal the same way, as $\cos(\cdot) = \dfrac{\exp(\mathrm{j}\cdot)+\exp(-\mathrm{j}\cdot)}{2}$. You mentioned you have audio experience; you almost certainly worked with real-valued data there. The frequency detection/estimation task then is in principle the same here, so this solution might be easier if you do not want to think about IQ-receivers and complex signals too much.
