My neighbor (on the other side of my backyard) bought their house couple years ago. They approach me and asked me they will have contractor to take off the brushes (on their side) between the boundary of two property and in next few days (4-8 days), they will install a fence. Our property has a bunch of evergreen trees next to the boundary line along with drainage stones. Personally, we don't want fence installed. Can I write an agreement to make sure the neighbor is responsible the all expenses for the fence including the installation, maintenance cost down the road and the agreement will need to carry to future property owners ? The property is in New Jersey.  Any suggestion is appreciated.

If they build inside the property line....you have no say so whatsoever.  If the fence, on their property, somehow damages the roots from your trees, that’s your problem.  Unless they actually damage something physically on your property, they can do whatever they want.

I agree with @Wayne Brooks. You have no say in the matter as long as they stay on their own property. At least they're not asking you to pay for 1/2 of it.

I agree with Wayne and John.  Your evergreens will cover or soften the view of the fence.