All palm trees flower. Most of us are used to seeing what looks like stringy yellow-ish-tan leaves around the fronds of a palm tree. It's the inflorescence, a cluster of flowers around a main stem. Many plants have this. On date palms, this is where the dates form. If you have a Phoenix dactylifera (a true date palm), cut the inflorescence off as soon as the flowers appear and it won't produce any of that messy fruit. Of course, if you want fruit, that's a different story. But most of us just want the plant to be ornamental.
Every year, I waiver back and forth on what to do with the inflorescences. I have tried to cut them off as soon as they appear, to keep the tree tidy, but that just makes things worse. They continue to grow out, and what's inside gets all brown and messy. Best to leave them on until they have fully extended and opened, and then cut them off after they flower. They will release a lot of tiny yellow flowers, which can be messy on a sidewalk, so keep an eye on it.
By the way, cycads don't flower at all. Your cycads (sago palms, etc.) won't do this.