Personally I think you have to try and track a quote down to its original source and that can be difficult. 'Brainyquote' is not great giving you the providence of an actual quote and I don't think a brainyquote link looks hectic in a PhD bibliography.
It it is such a well known quote you could try simply putting into Google and see what you get. Sometimes it's worth going beyond the first page on google, you can occasionally find a gem.
Alternatively try looking up the person who said it, as opposed to the quote itself. For quotes I often find consulting the hardcopy better, as in a book!! If you're near a library have a look for an original Oxford English Dictionary of Quotations and you might strike it lucky. I have four or five quotation books, given me by my late sister which I have found useful over time. If you want, give me the actual quote and who said it, and I will see if I can find it. Pm me if you would prefer.
I hate to be beaten by things like that!!