Styling ticks, axes and other elements in a Plot of a step function

For the arrow heads on the axes, use an Epilog inside the Plot with the Arrow function, or use the techniques described in this post: https://stackoverflow.com/questions/5844790/arrows-for-the-axes.

For the tick labeling, use for each item in the list of x-tick and y-tick locations not just the number but a list that includes the number and the corresponding label:

Ticks -> {{{-3, -3 Q}, {-2, -2 Q}, {-1, -Q}, {1, +Q}, {2, 2 Q}, {3, 3 Q}}, {-3, -2, -1, 1, 2, 3}} 

where I've done it just with the x-ticks. If you insist on having the "+" prefixes on the positive ones, you could change, say, {1,+Q} to:{1, TraditionalForm@HoldForm[+Q]} and similarly for the others.

Of course you could write a little function that you would do all that with the ticks simply by applying it to the list of x-tick numbers and the list of y-tick numbers.

By default, as you've seen, tick marks are drawn only to the positive side of the axis and at a predetermined length. To change that to get each tick mark crossing the axis, use an option third entry for each tick: a list {plen,nlen} giving (as a fraction of the image size, I believe) how far the tick mark should extend in the positive and negative direction from the axis. For example:

Ticks -> {{{-3, -3 Q, {0.01, 0.01}... 

You didn't say exactly what features of the displayed graphic you couldn't satisfactorily reproduce in Mathematica, but perhaps the size of the text, including tick labels, is an issue. In that case, you could use the BaseStyle-> option to Plot if you wanted to uniformly change all the text sizes, fonts, weights, etc. If, however, you want different treatment of different text elements, then you could modify each one by a Style treatment, e.g.:

Ticks -> {{{-3, -Style[3 Q, 24, Red, Bold, FontFamily -> "Papyrus"],... 

(My system has that font installed; yours may not.)

A function that works for any symmetric range:

plt[rng_] := Block[{r = rng, tcks, f}, f[x_] := Which[x == 1, "+Q", x == -1, "-Q", x > 0, "+" <> ToString[x] <> "Q", x < 0, ToString[x] <> "Q", x == 0, ""]; tcks = Transpose[{Range[-r, r], f /@ Range[-r, r] }]; Plot[Round[n], {n, -r, r}, Ticks -> {tcks, tcks}, AspectRatio -> Automatic, ExclusionsStyle -> Opacity[1], AxesLabel -> {"Input", "Output"}, AxesStyle -> Arrowheads[{-0.05, 0.05}], ImageSize -> 300]] plt /@ {3, 5} // Row 

I don't know what about the plot you would like to improve, but this would put a + in front of the positive tick marks and extend the ticks to the negative side of the axes

Plot[Round[n], {n, -3, 3}, AspectRatio -> Automatic, Exclusions -> None, AxesLabel -> {Input, Output}, Ticks -> Function[{xmin, xmax}, Table[{i, Row[{NumberForm[i, NumberSigns -> {"-", "+"}], "Q"}], {.02, .02}}, {i, Floor[xmin], Ceiling[xmax]}]] ] 

Slightly updated version in case you want +Q instead of +1Q

Plot[Round[n], {n, -3, 3}, AspectRatio -> Automatic, Exclusions -> None, AxesLabel -> {Input, Output}, Ticks -> Function[{xmin, xmax}, Table[{i, Row[{Switch[i, 1, "+", -1, "-", _, NumberForm[i, NumberSigns -> {"-", "+"}]], "Q"}], {.02, .02}}, {i, Floor[xmin], Ceiling[xmax]}] ] ] 

Final version incorporating the comment of the OP

Plot[Round[n], {n, -3, 3}, AspectRatio -> Automatic, Exclusions -> None, AxesLabel -> {Input, Output}, Ticks -> Function[{xmin, xmax}, Table[{i, Row[{ Switch[i, 1, "+", -1, "-", _, NumberForm[i, NumberSigns -> {"-", "+"}]], "Q"}], {.02, .02}}, {i, Floor[xmin], Ceiling[xmax]}]], AxesStyle -> With[{head = {Graphics[{Polygon[{{-1, 0.5`}, {0, 0}, {-1, -0.5`}}]}], 0.98`}}, Directive[Arrowheads[{{-0.03, 0, head}, {0.03, 1, head}}]] ] ] 

Something like this?

data = Table[{n - 1/2, n}, {n, -3, 3}]; ticks = {#, ToString[#] <> "Q"} & /@ {-2, -1, 1, 2}; ListLinePlot[ data, AspectRatio -> 1, PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, Ticks -> {ticks, ticks}, AxesLabel -> {"Input", "Output"}, DataRange -> {-3, 3}, InterpolationOrder -> 0 ]