How to draw one-line Venn diagram?

ref = StringSplit["A O I L D M E B C P K F N J H G Q"];

pos = PositionIndex[ref];

sorteddata = SortBy[pos@*First][SortBy[pos] /@ data];

Graphics[{Text[Style[#, 32], Append[pos@#, 0]] & /@ ref, 
  MapThread[{ Opacity[.5], #2, Disk[Append[Mean[pos /@ #], 0], {Length[#]/2, 1}]} &, 
    {sorteddata, ColorData[97] /@ Range[Length@data]}]}]

Alternatively,

Graphics[{Text[Style[#, FontSize -> Scaled[.05]], Append[pos@#, 0]] & /@ ref, 
  MapThread[{ Opacity[.5], #2, CapForm["Round"], 
     AbsoluteThickness[35], Line[Thread[{Flatten[MinMax[pos /@ #]], 0}]]} &, 
   {sorteddata, ColorData[97] /@ Range[Length@data]}]}, 
  ImageSize -> 600]

Here's one way, which wraps elements in nested colored Frames depending on their set membership. It's not the prettiest, but it's easy and handles non-contiguous intersections straightforwardly.

data = {{"A"}, {"J"}, {"Q"}, {"G", "H"}, {"I", "O"}, {"B", "C", "E", 
   "F", "K", "N", "P"}, {"B", "C", "D", "E", "F", "K", "L", "M", "P"}};

elements = "A O I L D M E B C P K F N J H G Q" // StringSplit;

colors = AssociationThread[
  data -> Map[ColorData[1], Range@Length@data]];

Row[KeyValueMap[
  Function[{element, memberships},
   Fold[Framed[#1, Background -> Lighter[#2], FrameStyle -> #2] &, 
    element, memberships]
   ],
  AssociationMap[Select[data, MemberQ[#]] & /* Map[colors], 
   elements]]];