No. At least I don't think so. Smaller monitor, less resources. Same goes for any variation, really. Lower FPS, less resources. Lower resolution, less resources. Same would go for size of the display.
Bigger screen almost always needs more power. Provided both are using same screen technology (TN, IPS, whatever else) and are about as old. (not brand new and five years old)
Power consumption is usually listed in display specs though.