Do quiet induction burners exist?
Quieter, yes.  Quiet, no.
This was based on my recent research where I tested out 8 different portable induction burners, did a lot of reading, and ultimately decided not to buy one.
Induction burners make noise for two main reasons:

They work via a spinning electromagnetic field, which causes both the burner and the vessel to vibrate microscopically, creating a loud hum.
They have an internal fan to keep the electronics cool while the burner is on.  This is often the louder source of noise.

Higher-end induction burners produce less noise by having a heavier case (reducing vibration), and by having a better cooling fan. However, having tried some of these fairly expensive commercial burners, like a Waring and a Vollrath, I'd say they are quieter but not exactly something you'd use in a room with a sleeping baby.
Built-in burners can reduce noise further simply by having sound insulation built into the cooktop, and many do.