Most effective way is to get your neighbour to cut off the inflorescence before the seeds mature and fall. They won't all grow into large plants, it'll be survival of the fittest. The seedlings will crowd each other out and most will die.

Thanks! I'll ask the neighbor to do that. Is that something that has to be done every year?

That's something that has to be done year round. Queen palms (syagrus romanzoffiana) will flower and fruit throughout the year. If the palm that is dropping the seeds is actually a queen palm, you should be able to pick up the seed/fruit when it falls, thus preventing that they sprout in your garden bed. Queen palm fruit is rather large (the size of large grapes) and each fruit contains a large seed that resembles a mini-coconut. Raking them up and disposing of them should be an easy task.

I do wonder if your neighbor has a Royal palm (roystonea), phoenix canariensis or washingtonia robusta. These have rather small seeds that can get easily lost in mulched garden beds and all three are rather prolific seeders. In any case, cutting off immature fruit or cutting off the inflorescence is the only way to stop them from producing viable seed. With any of these, it would take constant, year round monitoring.