Controller reported an aircraft experienced a pressurization problem requiring expeditious descent resulted in an airborne conflict.

Narrative

Aircraft X was in level flight at FL360 when they requested a descent to 15;000 due to a 'pressurization issue'. Aircraft Y was level at FL350 and if they would have stayed on their assigned routings they certainly would have lost lateral separation. I turned Aircraft Y 30 degrees right to make more space for Aircraft X to descend; I turned Aircraft X 30 degrees right as well and coordinated with Person 1; Person 2; and Person 3 so Aircraft X could descend to 15;000ft; and then land at ZZZ instead of going onto their original destination which was ZZZ1. Without reviewing the playback; I am not sure if they lost lateral separation with Aircraft Y or not. Though Aircraft X never declared an emergency; the urgency in their voice and the request to descend 21;000 feet; I treated them as an emergency aircraft. I'm not sure if any recommendations can be made for a case like this; but in the future I might just quick look the sectors impacted by the emergency depressurized aircraft and start making clearances then back coordinate afterward.