Electronic – NFC – design of inductor/“antenna” made of PCB traces

Concerning the fact that Q dropped when you used an impedance matching network:

You used a simple L matching network consisting of just two components (a capacitor and an inductance) which both are completely determined by the input and output impedance of your situation.
This leaves you no parameter to control Q. You get whatever Q is a result of the component values you need for matching.

So I suppose what you need is a more sophisticated matching network containing at least three components (e.g. a T- or a Pi-network) that has another degree of freedom which allows you to control not only input and output impedance but also Q.

Since you are also asking for references I recommend very much the chapter about impedance matching and Smith charts in Chris Bowick, "RF Circuit Design". It explains and contains an example problem for impedance matching while also caring about Q.

Have you read the following: Atmel AT02865 RF Layout with Microstrip ? It deals with exactly what you are doing, same chip and same balun.

One of the key parameter is the FR4 dielectric thickness between the RF ground plane and the microstrips. You have listed yours as 0.18mm (7 mils). In the Atmel app note, it uses an example of 0.25mm (10 mils) FR4 dielectric thickness. So the example is a little different, but is close enough that you can apply all the considerations while adjusting for the small dimensional differences accordingly.

Since the traces are sitting on top of a ground plane, you cannot use the coplanar waveguide model for the transmission lines. Using the microstrip model and dielectric thickness of 0.18mm, I get approx 0.3mm for 50-ohm line.

Also, you do want to isolate the microstrip lines from other elements of your design. For example, a quote directly from the Atmel app note page 10 -

For microstrip designs the copper pour on layer-1 should be kept away from the transmission line. The underlying Layer-2 ground plane needs to be the dominant ground reference to minimize variables. A keep-away distance of 4x the dielectric thickness will reduce the parasitic effects of copper pour to less than 1%. In other words the gap between the microstrip transmission line and copper pour on layer-1 should be 40 mils or more.

If you look at the balanced 100-ohm connection between the SAMR21 IC and the Balun, it ends up looking more like a "T" shape dipole antenna than a transmission line. I don't think there is anyway to make that better, so just make that connection as short as possible. Also for the same consideration of the preceding quote, I would probably take away that asymmetric ground pour.

Also, I would move the antenna to the left so the tail of antenna at the right is closer to the ground plane.