Whisk eggs, cream, salt, pepper, and nutmeg in a bowl. Stir in broccoli, onion, and Cheddar cheese. Pour egg mixture into the prepared souffle dish.
Turn on a multi-functional pressure cooker (such as Instant Pot(R)). Pour water into the pot and insert the trivet. Place the prepared dish on the trivet with an aluminum foil sling, folding its ends on top. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 25 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, about 10 minutes. Complete releasing pressure using the quick-release method, about 5 minutes. Open and remove the lid. Lift quiche by grabbing the ends of the foil sling. Let cool for 5 minutes.