No. You do not need an Xbox 360 Kinect for Xbox One. They are completely different, incompatible devices.
| Feature | Xbox 360 Kinect (v1) | Xbox One Kinect (v2) |
|---|---|---|
| Sensor | PrimeSense structured light (IR projector + CMOS) | Microsoft Time-of-Flight (ToF) IR + 1080p RGB |
| Resolution (Depth) | 320×240 @ 30Hz | 512×424 @ 30Hz |
| RGB Camera | 640×480 @ 30Hz | 1920×1080 @ 30Hz |
| Field of View | 57° H / 43° V | 70° H / 60° V |
| Skeletal Tracking | 20 joints, 2 players | 25 joints, 6 players |
| Connectivity | Proprietary USB + AC adapter | Proprietary wide USB 3.0 (no AC adapter) |
| Microphone Array | 4-mic | 4-mic (better beamforming) |
Buy the Xbox One Kinect (discontinued, used $40–$80 on eBay) or the Xbox One Kinect Adapter (for Xbox One S/X which lack the port) + Kinect sensor. No cross-gen workaround exists.