Yes, cold brew can speed up digestion and create the urge to poop. But it’s not just cold brew that will promote going #2; caffeine is responsible for sending you to the bathroom, and espresso - or any coffee drink for that matter - can have the same effect.
Healthline did a recent study on coffee and digestion, and the source found that caffeine “can activate contractions in your colon and intestinal muscles. Contractions in the colon push contents towards the rectum, which is the final section of your digestive tract.” This is what’s ultimately responsible for creating that urge.