Yes, closed convex surfaces in spaces of constant curvature are rigid. See Chapter V of Pogorelov's book [Extrinsic Geometry of Convex Surfaces][1]. 

As described at the beginning of the chapter on p. 270-271, the main idea  is to reduce the problem to the Euclidean case. Pogorelov says that this reduction is similar for elliptic and hyperbolic cases, and only considers the elliptic case, which is established in Thm. 1 on p. 321.


  [1]: https://www.ams.org/books/mmono/035/mmono035-endmatter.pdf