Strictly speaking, in AmE, a "marker" is a signed promise to pay a gambling debt. If a person placing a bet with a
bookie does not have cash available at the time, he might (if the bookmaker is amenable) sign a promissary note for the amount of money, which is called a "marker." If that person doesn't pay the money in a reasonable amount of time, the bookie might warn him "If you don't have the cash by Friday, I'm going to call in your marker," meaning that he will require the indebted person to do him some sort of favor (usually some unpleasant task - that's why he's relegating it to a person who basically has no choice).
Example: Fred owes Weasel the Bookie $500 for two unpaid markers. Every week Fred stops by and tells Weasel "I'll pay you next week, honest!" After several weeks of non-payment Weasel tells Fred: "I'm calling in your marker. Joe Blow has been romancing my girlfriend and he needs to be taught a lesson. I want you to go and break a few of his bones and tell him that Weasel said to stay away from his girl. You do this, and I'll call your debt paid in full."
Colloquially when someone "calls in a marker" he is asking for a very large favor from someone who "owes" him. But usually when folks have such favors owed them they like to use them sparingly and keep a few in reserve. In the OP's example, the speaker is not only worrying about using up all his owed favors in one fell swoop, he himself will have to offer similar favors to some people in order to extradite the prisoner. So some time in the future he will be called upon to give similar assistance to someone else; he will be indebted to them.
