You can't reset the millis() function, but you can count the amount of milliseconds and stop when necessary: beware though, the positioning might not be as precise as you wish and you have to consider the error will accumulate on the long run.

Nonetheless here is the code:

uint16_t duration;
bool direction = true;
void loop() {
  if (duration != 0 and !buttonPressed()) {
    // running back and forth if setup and no "finish" button detect
    uint16_t start = millis(); // reset the timer
    startMotor(direction);
    while (millis() - start < duration && !buttonPressed()) {
      ; // keep going
    }
    direction = !direction; // reverse direction
  } else if (duration == 0) {
    // your motor positioning setup
    while (!buttonPressed()) {
      ; // wait for button to start positioning
    }
    uint16_t start = millis(); // ask for time
    startMotor(true); // run the motor cw
    while (!buttonPressed()) {
      ; wait for button to end positioning
    }
    duration = millis() - start; // store duration
    direction = false; // reverse motor direction
    // next time loop() is executed the motor will go back to home
  }
}
bool buttonPressed() {} // define this to determine if the button has been pressed, safer if you use an interrupt
void startMotor(bool cw) {} // define this to run your motor clockwise or counterclockwise

Your code is not working because the millis is updating the sec variable.

After the sec reaches the 59 value, your if statement changes the sec to 0. However the millis will again put the latest time value i.e. 60 and lcd.print(sec) will print it. And the cycles continues, without ever resetting the sec value because millis is updating it.

One more thing, don't use delay(1000) to stop the controller for 1 second, You need make use of the sole purpose of the millis i.e. to avoid delays.

Here is your solution:

#include <LiquidCrystal.h>
LiquidCrystal lcd(4, 6, 10, 11, 12, 13);
long previousMillis = 0;
unsigned long sec = 0;
unsigned long mts = 0;
unsigned long hrs = 0;

void setup() {
    lcd.begin(16, 2);
}

void loop() {
    unsigned long currentMillis = millis();
    if(currentMillis - previousMillis > 1000) {
        previousMillis = currentMillis;   
        sec +=1;
        lcd.clear();
        lcd.setCursor(3, 0);
        lcd.print("Seconds:");
        lcd.setCursor(11, 0);
        lcd.print(sec);
        lcd.setCursor(7, 1);
        lcd.print("Mnt:");
        if (sec >= 59) {
            sec = 0;
            mts +=1;
        }
        lcd.setCursor(11,1);
        lcd.print(mts);
        }  
    }
}

In arduino UNO

extern volatile unsigned long timer0_millis;

noInterrupts ();
timer0_millis = 0;
interrupts ();

But it makes no sense at all.

Simply remember the previous millis and calculate time passed (by substracting).