Does Zhonya's Hourglass trigger a dying ready Riven, Broken Blade exhausting her?

Yes — because Zhonya's Hourglass exhausts her, and a ready unit that gets exhausted has "become exhausted."

Zhonya's isn't a triggered ability; it's a mandatory replacement effect that kills itself instead and performs its instructions — heal, exhaust, recall — as part of replacing the death (Rule 369.1, 373.1.a; FAQ #5836). Since Riven, Broken Blade is ready, the exhaust actually happens (Rule 415.1.b/c), so her "When I become exhausted" condition is met, the ability goes on the chain, and she can choose a mode that hasn't been chosen yet this turn — Buff me, spend a buff to Stun an enemy unit, or spend a buff to ready me. Note she'll be at Base by the time the trigger resolves (the recall is part of the same instruction), and the ready-me/Stun modes require her to already have a buff to spend, so usually the relevant line is "Buff me" now and ready her on a later exhaust.

One caveat: if she were already exhausted when the kill happened, nothing would happen on the exhaust (Rule 415.1.c), so no trigger.