What you want to do is change how the tiddler presents its content in view mode. But you don't want every tiddler to change how it appears.
Typically in this situation, you use a (real) template. Or actually, you modify the view template. To indicate that you want this tiddler treated differently, you typically do something like tag it with a special manner. Let's say you tag all quotes with "Quote". Although, if you never use "attribution" anywhere else, you could use the existence of that field. But for the moment, let's go with tag Quote.
Back up your TW, just in case. You might want to try this on TiddlyWiki.com just to see how the flow goes.
Create your tiddler with populated field "attribution". Put the text of the quote in the text box, because it's big and easy to edit. ;-) Put a tag "Quote" on the tiddler. Save it.
Edit the following tiddler (I'm assuming that you have a recent edition of TW -- I'm doing this based on v 5.1.19) :
$:/core/ui/ViewTemplate/body
Paste in the following contents and save the tiddler:
<$list filter="[all[current]!tag[Quote]]">
<$reveal tag="div" class="tc-tiddler-body" type="nomatch" stateTitle=<<folded-state>> text="hide" retain="yes" animate="yes">
<$list filter="[all[current]!has[plugin-type]!field:hide-body[yes]]">
<$transclude>
<$transclude tiddler="$:/language/MissingTiddler/Hint"/>
</$transclude>
</$list>
</$reveal>
</$list>
<$list filter="[tag[Quote]]">
<div style="background-color:black; color:white; padding:5px"><div align=left style="font-size: 200%">“</div><div style="margin:5px"><$view field="text"/></div><div align=right style="font-size: 200%">”</div><div style="font-size: 150%">-<$view field="attribution"/></div></div>
</$list>
Your tiddler should now appear formatted with text and attribution.
Note that what this does is wrap a <$list> structure around the original contents of the tiddler and displays them when the tag is not "Quote". Then it adds a 2nd <$list> to display your template text when the tag is "Quote" . So the lower part is the part you want to edit if you want to tweak your design. If you're using a recent version of TW, then the text should work out of the box.
Now any tiddler you tag with "Quote" will have it's text and attribution formatted per your template text.
HTH
-- Mark