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As a risk quantitative analyst at a sell-side CIB, based in a country with worlds most outstanding mathématicians, most of my colleagues never heard that kind of P/Q classification.
If you refer Q-quant as pure derivative pricing, stochastic analysis based market making, this kind of job is surely vanishing in traditional finance institutions.
This kind of work are more and more outsourced to finance software provider such as Numerix/Reuter/bloomberg, and the pricing functionality is already embedded in trading tools. Trading tools are increasingly automatics, so traders, FO quant no longer spend a whole week to implement a PDE kernel solver or debug a antithetic_WienerProcess_Generator_autroptr core dump. They just need to choose the Euler scheme from a scroll down list, entre Monte-Carlo path/sample number manually on a Murex interface, the trading tool immediately return different Net Present Value, according to your model parameter, discount curves etc.
Within a FO/MO quant teams, not everyone can resolve heston model by Fourier transform, nor could most of them distinguish the nuance between bridge pattern and adapter pattern, but almost every are capable of demonstrating BSM equation. That’s to say, the mathematical knowledge and programming skills might vary in an office, but every one should have a minimum understanding of stochastic calculus and sabr models character..... | 
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