  We compute the Hochschild cohomology of any block of $q$-Schur algebras. We
focus the even part of this Hochschild cohomology ring. To compute the
Hochschild cohomology of $q$-Schur algebras, we prove the following two
results: first, we construct two graded algebra surjections between the
Hochschild cohomologies of quasi-hereditary algebras because all $q$-Schur
algebras over a field are quasi-hereditary. Second, we give the graded algebra
isomorphism of Hochschild cohomologies by using a certain derived equivalence.
