Alberta has a legitimate claim, though 'world capital' is more marketing than geology. Here is the nuanced answer:
Alberta IS genuinely exceptional for dinosaurs. The Badlands region (around Drumheller and the Red Deer River Valley) contains some of the richest and most accessible dinosaur fossils on Earth. The Royal Tyrrell Museum is world-class, and Alberta has produced more named dinosaur species than most places globally. The rocks there are perfectly preserved Cretaceous period deposits — ideal for fossilization.
BUT — China, Mongolia, Argentina, and the western United States (especially Utah and Montana) have equally rich or richer fossil records. China alone has described more new dinosaur species in the last 20 years than Alberta has in its entire history. Mongolia's Gobi Desert is arguably as significant paleontologically. So 'capital' is a bit of provincial pride mixed with real scientific merit.
The honest take: Alberta is in the top tier globally for dinosaur paleontology. It has the infrastructure (Royal Tyrrell Museum), the accessibility (you can literally hike and see fossils in the badlands), and the scientific output to justify calling itself a world-leading dinosaur destination. But it is more accurate to say it is 'one of the world capitals' rather than THE capital.
Pro tip: If you are visiting Alberta for dinosaurs, the Royal Tyrrell Museum near Drumheller is genuinely world-class — it punches above its weight compared to larger museums. The self-guided badlands hikes are where you actually see fossils in situ, which is far cooler than any museum display.