Why does Glutamine have the symbol Q?
Initial letters of the names of the amino acids were chosen where
  there was no ambiguity. There are six such cases: cysteine, histidine.
  isoleucine, methionine, serine and valine. All the other amino acids
  share the initial letters A, G, L, P or T, so arbitrary assignments
  were made. These letters were assigned to the most frequently
  occurring and structurally most simple of the amino acids with these
  initials, alanine (A), glycine (G), leucine (L), proline (P) and
  threonine (T).
  
  Other assignments were made on the basis of associations that might be
  helpful in remembering the code, e.g. the phonetic associations of F
  for phenylalanine and R for arginine. For tryptophan the double ring
  of the molecule is associated with the bulky letter W. The letters N
  and Q were assigned to asparagine and glutamine respectively; D and E
  to aspartic and glutamic acids respectively. K and Y were chosen for
  the two remaining amino acids, lysine and tyrosine, because, of the
  few remaining letters, they were close alphabetically to the initial
  letters of the names. U and O were avoided because U is easily
  confused with V in handwritten material, and O with G, Q, C and D in
  imperfect computer print-outs, and also with zero. J was avoided
  because it is absent from several languages.
  
  Two other symbols are often necessary in partly determined sequences,
  so B was assigned to aspartic acid or asparagine when these have not
  been distinguished; Z was similarly assigned to glutamic acid or
  glutamine. X means that the identity of an amino acid is undetermined,
  or that the amino acid is atypical. "Nomenclature and Symbolism for Amino Acids and Peptides"