When did countries start using passports?

Ancient times. Biblical era in the middle east. Used by the early Chinese in that part of the world.

They became more frequently used in the Middle Ages in Europe

And of course they are ubiquitous today.

Are there Biblical references to passports required for people to travel?

Outside China or the Middle East, I (with Wikipedia assistance – https://en.wikipedia.org/wiki/Passport#History) don’t know of an example of something like them being used before the Middle Ages, and then more as visas than passports, before Henry V.

Marco Polo had to have papers?