If you are still checking this, I think your goal is plausible with some heavy autodidatics. The poster who mentioned that no one will look at your transcript out of academia is correct IMHO but since a PhD is often regarded as 'super-education' you could possibly learn enough on your own and if you are able to make known your expertise to financial employers, they may very will value you more than relevant majored MS's due to the three letters after your name (I know this as a MS Stats person myself :) ).
As another poster mentioned, knowledge of stochastic modeling is very often expected. Steven Shreve's "Stochastic Calculus for Finance II: Continuous Time Models" is the text we use. There may be a less mathematical text that would still do the trick (there are several on the topic, so try to find one that you can learn from ; the subject is considered difficult even by mathematicians).
What are generally considered precursors to Stochastic Calculus are multivariate calculus, partial differential equations and some basic real analysis as would be taken by a Math undergrad.
Numerical Analysis may or may not be very important for what you actually do on the job, but in my experience in an interview you're more likely to be asked about Stochastics.
Don't be too intimidated by this subject matter, it is learnable. You may justifiably think you do not need to know this kind of thing to contribute to a quantitative role in some way and you would be right. However, the field of finance was greatly influenced by the amount of physicists who became quants in the last 30 or so years. As a result, to work in quantitative finance, you are expected to know advanced math (or be able to fake it pretty well).
One other headwind is that traditional quant jobs are becoming less prevalent due to financial regulations and other industry trends. One upshot for you is that risk management is becoming a hiring area at financial firms. Risk management is generally considered to be more statistical however I personally intern in this area and I am expected to be able recognize financial mathematics terms of art (such as Black Scholes with is a famous stochastic differential equation model). The job you get may end up being largely econometrics and statistics but there is a barrier to entry (which I do think you would be able to surmount with some hard work).
Regarding statistics and econometrics, it might be also helpful for you to know that in Finance the type of data people are interested in analyzing is almost uniformly time series. The standard text in this regard is Ruey Tsay's "Analysis of Financial Time Series". In terms of the actual subject matter of Finance as it relates to statistics, you'd benefit from reading up on Markowitz portfolio theory. There are several quantitative portfolio texts and I don't have a favorite, but this knowledge would also be key in risk management.
Additionally, quants are almost uniformly expected to be comfortable with object oriented programming (such as C++, C# and Java). Knowledge of things like Matlab or R is expected but also considered learnable on the job while object oriented programming is a prerequisite. A pretty good C++ book that also covers the object oriented paradigm is Deitel and Deitel. Or you could just start writing code and learn by experimentation which is usually more effective (and fun :) ).
Finally, if you have not come across, Mark Joshi wrote a piece for 'wannabe quants' here:http://www.markjoshi.com/downloads/advice.pdf
Best of luck to you.