I don't think you want a mechanic that forces actions and takes away player agency, not one bit. People do squelch on their debts.
What DWeird said is on the right track, I think. It's about leverage, and what happens to your rep and your standing in this favor-based economy if you squelch. Also, is there an internal component to it? Like, are honorable characters more compelled to act on a debt? If so, do they get anything out of being honorable?
This mostly gets into what Vincent's always talking about regarding your insights. How do *you* think owing someone a favor affects your behavior? Or, how do you think it affects the kind of person your hack is about? How do you think squelching affects the overall economy of favors that seems to be important to your game? And how does that influence the decisions of the players (or the PCs)?
Then translate that believe into moves & mechanics. You've got three cases to address, right?
* PCs calling an NPC's marker
* NPCs calling a PC's marker
* PCs calling a different PCs marker
I'd probably go with something like:
* Keep a list of favors you owe people, and favors they owe you.
* If someone owes you, you can use that debt as leverage to manipulate them
* If word gets out that squelched on a debt, then... (one of these? all of these? the MC chooses?)
**People stop taking your calls
**Someone comes looking for you
**When you manipulate an NPC, treat a 10+ as a 7-9. Your word ain't worth shit.
Maybe have a playbook move like Man of Your Word. If an NPC knows your rep, treat a 7-9 as a 10+ when you manipulate them. If you ever squelch on a debt, cross this move off.