A single transistor can have a voltage gain as high as about 180 and it amplifies its own noise. Its noise is very low so its output level of the noise is very low and will not increase its power consumption.
A power amplifier can amplify mains hum noise and produce a high output power to a speaker. Then its power consumption is high.
Low noise circuits like preamps typically use larger quiescent current, because noise roughly rises as the impedance rises.
But the main question is noise WHERE? and power consumption of WHAT circuit?
For example electrical noise on primary produced by a switching converter could rise with load power.