exp = Expectation[
  Abs[x] Abs[y], {x, y} [Distributed]
        MultinormalDistribution[{0, 0}, {{a, c}, {c, b}}]];

exp =
   exp /. ArcTan[x_] -> Pi/2 - ArcTan[1/x] //FullSimplify[#, c > 0] & //
   Collect[#, Sqrt[_], FullSimplify] & // (# /.
   ArcCot[x_] -> Pi/2 - ArcCot[1/x]) & // Simplify

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