The answer is NO, changing the PWM divider for pin 11 will not affect the millis() function (nor delay()).
Explanation:
The Arduino Nano uses the ATMega328P microcontroller, same as the Arduino Uno.
Pin 11's PWM is controlled by Timer 2. By default Timer 2 has a prescale of 64 (ie. setting 0x04). The base frequency of Timer 2 is 31372.55 Hz resulting in a frequency of 490.2 Hz (ie. 31372.55/64 = 490.1961).
On the other hand, the microsecond timer (ie. the one used by millis(), micros(), delay() and delayMicroseconds()) is controlled by Timer 0.
So changing the prescaler of Timer 2 will not affect millis().
It will however change the PWM frequency of Pin 3.
Back to your question:
The possible PWM frequencies of Pin 11 (higher than 488 Hz) are:
31373 / 32 = 980.4 Hz
31373 / 8 = 3921.6 Hz
31373 / 1 = 31373 Hz
Where 32, 8, 1 are the prescaler.
Which equates to a setting value of 0x03, 0x02 & 0x01 respectively.
Add this line of code:
TCCR2B = TCCR2B & 0b11111000 | setting;
Where setting is the value of the setting for the respective prescaler.
============================================
|| Frequency [Hz] || Prescaler || Setting ||
============================================
|| 31373.55 || 1 || 0x01 ||
|| 3921.57 || 8 || 0x02 ||
|| 980.39 || 32 || 0x03 ||
|| 490.20 || 64 || 0x04 ||
|| 245.10 || 128 || 0x05 ||
|| 122.55 || 256 || 0x06 ||
|| 30.64 || 1024 || 0x07 ||
============================================
Source code: https://github.com/arduino/ArduinoCore-avr/blob/master/cores/arduino/wiring.c