The simple answer is NO, the Starlink router does not have a WPS button
To be honest, if you need a WPS button to setup your device, it means the device you are linking to is probably a cheap printer or cheap Wi-Fi booster.
Better quality printers will have an interface where you can select the Wi-fi network and enter a password, usually from a LCD display on the printer.
The more expensive Brother printers usually have this facility. Cheaper Brother, HP , Epson and Canon printers do on have this facility.
Our advise is to avoid cheaper devices that require a WPS button to pair your Starlink wi-fi.
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