How to represent the following system function in a block diagram? I am aware that this a simple question, but I haven't been able to find a single reference that explains this situation. The system function is:

$$\frac{1}{1-(0.3)(1+j)D}$$

How can one draw a block diagram with only real components to represent something that is complex?

The system function is

$$\frac{Y}{x}  = \frac{1}{1-(0.3)(1+j)D} \implies Y(1-(0.3)(1+j)D)=X$$
where $Y$ is the output and $X$ is the input.

Therefore $$Y-DY(0.3)(1+j)=X \implies y[n]=0.3y[n-1](1+j)+x[n]$$
It is a simple first-order recursive system like this:



In your case $b[0]=1$, $b[1]=0$, and $a[1]=-0.3(1+j)$. Of course the coefficient is complex if your system is complex. What you can do is split the real and imaginary parts and get two real-valued systems. The output of one is the real part of the output signal $y[n]$, and the output of the other is the imaginary part:

$$y_R[n]=0.3y_R[n-1]-0.3y_I[n-1]+x_R[n]\\
y_I[n]=0.3y_R[n-1]+0.3y_I[n-1]+x_I[n]$$

where $y_R[n]$ and $y_I[n]$ are the real and imaginary parts of $y[n]$, respectively (and the same for $x[n]$). If the input signal is real-valued then of course $x_I[n]=0$.

As for general complex-valued systems also have a look at this figure from the textbook Digital Communication by Lee and Messerschmitt: