Do I need to pay a monthly fee for the apple watch

And if you decide not to use the cellular radio, or you get a GPS only Apple Watch, then there are no fees.

A GPS only Apple Watch, can do everything an activated cellular Apple Watch can do as long as your iPhone is nearby. Where nearby is somewhere in the same moderately sized home.

If you have a GPS+Cellular  Watch and want to use the cellular capabilities, then your iPhone service provider has to offer support for the watch and will add $10 to $15 to your monthly phone bill for the watch support.

M1 Mac mini, macOS 13.3.1;  Watch, series 7 GPS/cellular, watchOS 9.4; iPad Pro 2nd Gen, iPadOS 16.4.1; iPhone 12 Pro, iOS 16.4.1; TV 4th Gen, tvOS 16.4.1; HomePods, audioOS 16.4.1 and Airport Express w/ Airplay 2