Yes! Well, kind of.
If you're currently typing a quote while on Typing Practice, click on the title of the quote and it will take you to the quote's page. Then, if you choose to "Train on this quote," you can type that same quote as many times as you desire. However, the WPM/accuracy that you reached on that quote will not be factored into your overall accuracy and WPM.
The only thing is, you can't "save" this quote- you do have to keep on searching for the same one. The only real way to save a quote is to write your own, and I wouldn't recommend copy+pasting someone else's quote into the publisher, since that wouldn't be very great to the other typists, and definitely not great to the person who submitted that quote.
So, all in all, you can continue to use a quote for typing, but you can't exactly save that quote unless you write it.
Hope I helped!

There is also the search field. You can take some keywords specific to your precious quote and go for it.