to bill and coo

from The Collaborative International Dictionary of English v.0.48

Bill \Bill\, v. i. [imp. & p. p. {Billed}; p. pr. & vb. n.
   {Billing}.]
   1. To strike; to peck. [Obs.]
      [1913 Webster]

   2. To join bills, as doves; to caress in fondness. "As
      pigeons bill." --Shak.
      [1913 Webster]

   {To bill and coo}, to interchange caresses; -- said of doves;
      also of demonstrative lovers. --Thackeray.
      [1913 Webster]