millis() and micros() overflow periodically. However, this is not a problem: as long as you compare durations instead of timestamps you can forget about the overflows.

The liked answer also gives the trick for resetting millis(). Can be handy for testing purposes, but you do not need this to handle the millis() rollover problem. Notice that there is no clean way to reset micros(): it relies on a static variable, i.e. a variable private to the file defining it.

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);
        }  
    }
}

You don't. Instead you just remember what millis() was when you pressed your start button, then subtract that from whatever millis() is showing at any other point in the future.