I wanted to know if this increase in frequency would cause internal noise to appear on the analog pins.
Depends on what you mean by "internal noise". Increasing the ADC frequency does lower the accuracy of the conversion, so of the 10 bit you get returned only the upper 6-8 bits are usable. If that's noise for you, then you get noise.
You always get non-precise results if you read different analog pins in high frequency because the ATmega has only one ADC and an analog MUX that connects the different analog pins to the ADC. After switching to another pin the ADC needs some time till it provides an accurate result (the internal capacity needs to adapt to the new voltage).