Probably should be moved here:
SpaceX
To address your question though no that won't work.
That's a basic install video- but the cable from the dish also supplies power, and runs from dish to power supply ethernet at both ends.... then another port on the power supply connects it to the router.
Actually I think there MIGHT be a way you could technically hook it up, but it'd be goofy as hell and no idea how well (if at all) it'd work.
Your issue is the Starlink power supply really since it wants ethernet in and out and supplies power out on both ports (to the router and dish respectively).
So:
Coming FROM the dish in theory you could daisy chain a POE injector to the dish to give it power, then connect an Ethernet to MOCA adapter to the other injector port to bridge to coax, then a MOCA to ethernet on the other end you would plug into...another POE injector, the other port going to the Starlink router.
Removing the Starlink power supply entirely from the system and letting each POE injector power the dish and router.
You can check with starlink, but I'd be mildly surprised if they were willing to support such a config even if you get it working... and I can't imagine buying 2 injectors and 2 MOCA adapters would actually be any cheaper than just running the 1 ethernet cable that comes with the system into the house.