Gdddk

One possibility: instead of a tree, I just used some coordinates and decorated paths; I used the triangle 45 arrow tip from the arrows library, changed the gluon style to use decoration=coil,aspect=0, and suppressed parts of the original code not relevant here:

documentclass[11pt]article
usepackagetikz
usetikzlibrarypositioning,arrows
usetikzlibrarydecorations.pathmorphing
usetikzlibrarydecorations.markings

begindocument

tikzset
particle/.style=thick,draw=blue, postaction=decorate,
 decoration=markings,mark=at position .5 with arrow[blue]triangle 45,
gluon/.style=decorate, draw=black,
 decoration=coil,aspect=0
 

begintikzpicture[node distance=1cm and 1.5cm]
coordinate[label=left:$e^-$] (e1);
coordinate[below right=of e1] (aux1);
coordinate[above right=of aux1,label=right:$e^-$] (e2);
coordinate[below=1.25cm of aux1] (aux2);
coordinate[below left=of aux2,label=left:$e^-$] (e3);
coordinate[below right=of aux2,label=right:$e^-$] (e4);

draw[particle] (e1) -- (aux1);
draw[particle] (aux1) -- (e2);
draw[particle] (e3) -- (aux2);
draw[particle] (aux2) -- (e4);
draw[gluon] (aux1) -- node[label=right:$gamma$] (aux2);
endtikzpicture
enddocument

After an edit to the question, here's some new code with the required coil decoration:

documentclass[11pt]article
usepackagetikz
usetikzlibrarypositioning,arrows
usetikzlibrarydecorations.pathmorphing
usetikzlibrarydecorations.markings


begindocument
tikzset
particle/.style=thick,draw=blue, postaction=decorate,
 decoration=markings,mark=at position .5 with arrow[blue]triangle 45,
gluon/.style=decorate, draw=black,
 decoration=coil,aspect=0.3,segment length=3pt,amplitude=3pt
 

begintikzpicture[node distance=1cm and 1.5cm]
coordinate[label=left:$G$] (e1);
coordinate[below right=of e1] (aux1);
coordinate[above right=of aux1,label=right:$N$] (e2);
coordinate[below=1.25cm of aux1] (aux2);
coordinate[below left=of aux2,label=left:$N$] (e3);
coordinate[below right=of aux2,label=right:$G$] (e4);

draw[gluon] (e1) -- (aux1);
draw[particle] (aux1) -- (e2);
draw[particle] (e3) -- (aux2);
draw[gluon] (aux2) -- (e4);
draw[particle] (aux2) -- node[label=right:$N$] (aux1);
endtikzpicture
enddocument

You can also use the feynMP package to create Feynman diagrams. It uses metapost instead of tikz, but provides a nice interface for creating different types of Feynman diagrams. For example:

documentclassarticle
usepackagefeynmp
DeclareGraphicsRule*mps*

begindocument
unitlength = 1mm
beginfmffilegluon
beginfmfgraph*(40,30) 
 fmflefti1,i2 
 fmflabel$e^-$i1
 fmflabel$e^-$i2
 fmfrighto1,o2
 fmflabel$e^-$o1
 fmflabel$e^-$o2
 fmffermion, fore=bluei1,v1,o1 
 fmffermion, fore=bluei2,v2,o2
 fmfphoton,label=$gamma$v1,v2 
 fmfdotv1,v2
endfmfgraph*
endfmffile
write18mpost gluon
enddocument

Run this twice with pdflatex --shell-escape filename, and you will get

These diagrams are very easy to produce with the new tikz-feynman package (see also the project page), based on this answer. The only requirement for using automatic positioning of the vertices is to compile the document with lualatex:

documentclass[tikz]standalone
usepackage[compat=1.1.0]tikz-feynman
begindocument
feynmandiagram [vertical=b to a] 
 i1 [particle=(textupe^-)] -- [fermion] a -- [fermion] i2 [particle=(textupe^-)],
 a -- [photon,edge label'=(gamma)] b,
 f1 [particle=(textupe^-)] -- [fermion] b -- [fermion] f2 [particle=(textupe^-)],
;

tikzfeynmanset
 every fermion=draw=blue,

feynmandiagram [vertical=b to a] 
 i1 [particle=(N)] -- [fermion] a,
 i2 [particle=(G)] -- [photon] a,
 a -- [fermion,edge label'=(N)] b,
 b -- [photon] f1 [particle=(G)],
 b -- [fermion] f2 [particle=(N)],
;
enddocument

Here is my take on this. I think it produces beautiful output and doesn't require running any special programs like feynmp. Drawback: you have to put in the nodes manually, but that's very managable imho. Bonus: it retains full flexibility for using any kind of tikz decorations and so on while at the same time making it easy to use due to the tree approach.

documentclassstandalone

usepackage[utf8]inputenc
usepackagetikz
usetikzlibrarytrees
usetikzlibrarydecorations.pathmorphing
usetikzlibrarydecorations.markings

begindocument

% Define styles for the different kind of edges in a Feynman diagram
tikzset
 gageboson/.style=decorate,decoration=snake,draw=tud9a,
 electronin/.style=draw=tud1a,postaction=decorate,decoration=markings,mark=at position .55 with arrow[draw=tud1a]<,
 electronout/.style=draw=tud1a,postaction=decorate,decoration=markings,mark=at position .55 with arrow[draw=tud1a]>,
 gluon/.style=decorate,draw=magenta,decoration=coil,amplitude=4pt,segment length=5pt


begintikzpicture[thick,
 % Set the overall layout of the tree
 level/.style=level distance=1.2cm,
 level 2/.style=sibling distance=2.6cm,
 level 3/.style=sibling distance=2cm
 ]
 coordinate
 child[grow=up]
 child 
 node $k^'_1$
 edge from parent [electronout]
 
 child 
 node $k_1$
 edge from parent [electronin]
 
 edge from parent [gageboson] node [right=3pt] $q$
 
 child[grow=down,level distance=0pt]
 child 
 node $k_2$
 edge from parent [electronin]
 
 child 
 node $k^'_2$
 edge from parent [electronout]
 
 ;
 fill[black] (0,0) circle (.05cm);
 fill[black] (0,1.2cm) circle (.05cm);
endtikzpicture

enddocument

Yields:

However, when you look closely, the wriggly line of the phonon is not perfect. Anybody know how to get rid of that linear part?

In the particular case asked above, the code would be:

documentclassstandalone

usepackage[utf8]inputenc
usepackagetikz
usetikzlibrarytrees
usetikzlibrarydecorations.pathmorphing
usetikzlibrarydecorations.markings

begindocument

% Define styles for the different kind of edges in a Feynman diagram
tikzset
 gageboson/.style=decorate,decoration=snake,draw=black,
 electronin/.style=draw=tud1a,postaction=decorate,decoration=markings,mark=at position .55 with arrow[draw=red]<,
 electronout/.style=draw=tud1a,postaction=decorate,decoration=markings,mark=at position .55 with arrow[draw=red]>,
 gluon/.style=decorate,draw=magenta,decoration=coil,amplitude=4pt,segment length=5pt


begintikzpicture[
 thick,
 % Set the overall layout of the tree
 level/.style=level distance=1.5cm,
 level 2/.style=sibling distance=2.6cm,
 level 3/.style=sibling distance=2cm
 ]
 coordinate
 child[grow=up]
 child 
 node $N$
 edge from parent [electronout]
 
 child 
 node $G$
 edge from parent [gageboson]
 
 edge from parent [electronout] node [right=3pt] $N$
 
 child[grow=down,level distance=0pt]
 child 
 node $N$
 edge from parent [electronin]
 
 child 
 node $G$
 edge from parent [gageboson]
 
 ;
 fill[black] (0,0) circle (.05cm);
 fill[black] (0,1.5cm) circle (.05cm);
endtikzpicture

enddocument

and results in this