Can trees be made into graphs?

You can use halirutan's makeTree function from this answer. Your purpose is slightly different than the purpose in that question, so the function can be simplified a bit in this context:

makeTree[nodes_] := Module[{counter = 0},
  traverse[h_[childs___]] := With[{id = counter},
    {DirectedEdge[id, ++counter], traverse[#]} & /@ {childs}
    ];
  traverse[_] := Sequence[];
  TreeGraph[#, GraphLayout -> "LayeredDigraphEmbedding"] &@Flatten[traverse[nodes]]
  ]

Use it like this:

expr = TreeForm[{a, {a, {a, a, a}}}];
{expr, makeTree @@ expr}

TreeFormToGraph[treeForm_] := 
 Module[{tree = ToExpression@ToBoxes@treeForm, order, pos, label},
  label = Cases[tree, Inset[name_, n_] :> Rule[n, Placed[name, Center]],Infinity];
  {order, pos} = Catenate /@ Cases[tree, 
     Line[order_] | GraphicsComplex[pos_, ___] :> {order, pos}, Infinity];
  Graph[UndirectedEdge @@@ order, VertexLabels -> label, 
   VertexCoordinates -> MapIndexed[Rule[First[#2], #] &, pos]]]

Note the result of TreeFormToGraph is Graph object.

Example 1:

Example 2:

Update: We can use GraphComputation`ExpressionGraph to get a one-liner that converts a TreeForm object to a Graph object:

treeFormToGraph = Apply[GraphComputation`ExpressionGraph];

treeFormToGraph @ TreeForm[{{{a,b},c},d}]

We can add styling to get a Graph that looks like TreeForm:

ClearAll[treeFormToGraph ]
treeFormToGraph[t_TreeForm, o : OptionsPattern[]] := 
   Module[{g = GraphComputation`ExpressionGraph[t[[1]], o, 
           VertexSize -> {"Scaled", .1}, VertexStyle -> LightYellow, 
           VertexShapeFunction -> "Rectangle"]}, 
      SetProperty[g, VertexLabels -> (PropertyValue[g, VertexLabels] /. 
              Rule[a_, b_] :> Rule[a, Placed[b, Center]])]];

treeFormToGraph[TreeForm[{{{a,b},c},d}], VertexStyle->Pink]

Original answer:

We can use, instead of TreeForm, GraphComputation`ExpressionGraph which produces a Graph object accepting all the options of Graph.

g1 = GraphComputation`ExpressionGraph[{{{a, b}, c}, d}, 
   VertexSize -> {"Scaled", .1}, VertexStyle -> LightYellow, 
   VertexShapeFunction -> "Rectangle"];

SetProperty[g1, VertexLabels -> (PropertyValue[g1, VertexLabels] /. 
    Rule[a_, b_] :> Rule[a, Placed[b, Center]])]