Since the required object will no longer be a tree (it has cycles), then it could be better to draw it without tikz-qtree
; the following code shows some possibilities:
\documentclass[landscape]{article}
\usepackage{tikz}
\usetikzlibrary{positioning,shadows}
\begin{document}
\tikzset{font=\small,
every node/.style=
{top color=white,
bottom color=blue!25,
rectangle,rounded corners,
minimum height=8mm,
draw=blue!75,
very thick,
drop shadow,
align=center,
text depth = 0pt
},
myedge/.style=
{draw=blue!50,
thick
}}
\begin{tikzpicture}
\node (a) {A};
\node[below=of a] (d) {D};
\node[right=of d] (e) {E};
\node[left=of d] (b) {B};
\node[below=of b] (c) {C};
\foreach \pare/\chil in {a/b,a/d,a/e,b/c,d/c}
\draw[myedge] (\pare.south) -- (\chil.north);
\draw[myedge] (a.south) -- (c.north);
\begin{scope}[xshift=5.5cm]
\node (a) {A};
\node[below=of a] (d) {D};
\node[right=of d] (e) {E};
\node[left=of d] (b) {B};
\node[below=of b] (c) {C};
\foreach \pare/\chil in {a/b,a/d,a/e,b/c,d/c}
\draw[myedge] (\pare.south) -- (\chil.north);
\draw[myedge] (a.south) .. controls (-3.5cm,-1.5cm) .. (c.north);
\end{scope}
\begin{scope}[xshift=8cm]
\node (a) {A};
\coordinate[below=of a] (aux);
\node[right=of aux] (b) {B};
\node[right=of b] (d) {D};
\node[right=of d] (e) {E};
\node[below=of aux] (c) {C};
\foreach \pare/\chil in {a/b,a/d,a/e,b/c,d/c}
\draw[myedge] (\pare.south) -- (\chil.north);
\draw[myedge] (a) -- (c);
\end{scope}
\end{tikzpicture}
\end{document}

If you want to keep the tikz-qtree
syntax and style, then you could do something like this:
\documentclass[landscape]{article}
\usepackage{tikz-qtree}
\usetikzlibrary{shadows,trees,calc}
\begin{document}
\tikzset{font=\small,
edge from parent fork down,
level distance=1.75cm,
every node/.style=
{top color=white,
bottom color=blue!25,
rectangle,rounded corners,
minimum height=8mm,
draw=blue!75,
very thick,
drop shadow,
align=center,
text depth = 0pt
},
edge from parent/.style=
{draw=blue!50,
thick
}}
\begin{tikzpicture}
\Tree [.\node (a) {A};
[.{B}
[.\node (c) {C}; ]
]
[.\node (d) {D};
]
[.{E}
]
]
\draw[edge from parent] let \p1=(d.south), \p2=(c.north) in
(d.south) |- ( $ (d.south) + (0,0.5*\y2-0.5*\y1) $ ) -| (c.north);
\draw[edge from parent] let \p1=(a.south), \p2=(d.north) in
(a.south) -- ++(0,0.5*\y2-0.5*\y1) -- ++(-1.3,0) |- ( $ (c.north) - (0,0.5*\y2-0.5*\y1) $ ) -- (c.north);
\end{tikzpicture}
\end{document}

tikz
primitives to draw it (I don't knowtikz
.)dot
component of Graphviz. See this question for embedding it in a LaTeX document.