Pokémon Quest and all other free-to-play Nintendo Switch games will not require a paid Nintendo Switch Online membership when the new service arrives next month. The following info comes from ResetEra user ggx2ac:
If you go to the eShop, and look at an online game like Minecraft, it will have something like this in the fine print:
“To use online services you must have access to a wireless internet connection. Some online services may not be available in all countries. From the full launch of the Nintendo Switch Online membership service in 2018, online play requires a paid membership.”
Lots of retail games with online play have that fine print.
However, if you look at the following free to play games:
Paladins
Fortnite
Pokémon Quest
Dawn of the Breakers
Fallout Shelter
Galak-Z: Variant S
Zaccaria Pinball
The Pinball Arcade
Kitten Squad
Pinball FX3
None of them have that fine print.
Source: ResetEra.com