I got a MacBook Air zu for Christmas and now I wanted to ask if you can connect the PS4 to the MacBook. That I can play on the MacBook. Do you need a specific cable or system on the MacBook?
No, you can't
You just need an adapter for the Mac that has Hdmi.
Do you know what such an adapter is called?
That goes over https://remoteplay.dl.playstation.net/remoteplay/lang/de/index.html. The program streams your PS4 to your MacBook and can also be controlled via your MacBook if you connect the controller to it (via USB cable or a Bluetooth adapter). At least that's the case with Windows, I don't know exactly how it is with Apple products.
As far as I know, this only works on Windows. However, there are programs with which you can install Windows on your Mac - so that should also work with your PS4 - but it costs money ;-)
Does your mac only have thunderbolt or also usb-a?
If he has usb c then the HOTUCG USB C Hub, 6 IN 1 USB C Adapter USB-C charging port (Thunderbolt 3), SD / Micro SD card reader, 3 USB 3.0 ports for MacBook Pro 2019/2018/2017/2016, MacBook Air 2019/2018, (space gray) https://www.amazon.de/...07Y9K28SR/
This can be done wirelessly via PS4 Remote Play
But for me the quality is bad and it often depends on what should I do about it?
As for the quality, you can have a look in the settings. There you can set the qualification to 720p or 1080p. Regarding the hangers, however, I can't help you, since that is mostly due to poor internet.