exp = Expectation[
  Abs[x] Abs[y], {x, y} [Distributed]
        MultinormalDistribution[{0, 0}, {{a, c}, {c, b}}]]// FullSimplify[#, c > 0] &;

expN =
  Expectation[
   Abs[x] Abs[y], {x, y} [Distributed]
   MultinormalDistribution[{0, 0}, {{1, 1/4}, {1/4, 2}}]] // N