Heat a 4-quart saucepan over medium heat. Add oil to the pan; swirl to coat.
Add onion, zucchini, and carrot; cook, stirring constantly for 5 minutes.
Add garlic; cook, stirring constantly for 1 minute. Stir in the salt, pepper and tomatoes; bring to a simmer, and cook for 5 minutes, stirring occasionally.
Coat the inner pot Instant Pot with cooking spray.
Combine basil, ricotta, spinach and goat cheese in a medium bowl.
Spread 1/2 cup spinach mixture in the cooker, Arrange 1/3 of the lasagna noodles over spinach mixture in the pot, breaking noodles as necessary to fit in the pot; top with half of the remaining spinach mixture and 1 cup tomato mixture. Repeat procedure once, ending with noodles.
Pour the remaining tomato mixture over noodles, being careful to cover noodles completely.
Close and lock the lid of the Instant Pot. Turn the steam release handle to "Venting" position. Press Slow Cook function and adjust to select "less" mode. Adjust time to 2 hours cooking.
When time is up, uncover and sprinkle with Parmesan cheese; cover and let stand 15 minutes before serving. Garnish with basil leaves, if desired.