Turn on a multi-functional pressure cooker (such as Instant Pot) and select the sauté function. Heat olive oil until hot, then add mushrooms, onion, celery, carrot, and garlic. Cook, occasionally stirring, until vegetables soften, about 3 minutes. Turn the pressure cooker off.
Stir in chicken stock, wild rice, wine, thyme, rosemary, salt, and pepper. Close and lock the lid and set the pressure valve to sealing. Select high pressure according to manufacturer's instructions; set timer for 30 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to the manufacturer's instructions, for 15 minutes. Then release the remaining pressure carefully using the quick-release method according to the manufacturer's instructions, about 5 minutes. Unlock and remove the lid.