Would the battery from a 13" MacBook Pro (2010) work in a 15" MacBook Pro 2011?
Nope. They're different sizes. Compare iFixit's guide to replacing a 15" early 2011 battery to their guide for a 2010 13":

15": 



13":



Both batteries are roughly proportional to their cases, and the 15" has a substantially bigger case than the 13". Additionally, they're shaped differently - see the little notch in the top left of the 15" battery, which isn't there on the 13" battery. It physically won't fit.