Note there's broadly two kinds of quants.
One is a derivatives quant, basically someone who spends a lot of time working out the values of derivatives. So you're looking at ito calculus and iterative methods to find the value of contracts with weird clauses (Knockouts, Cliquets, vanillas, etc), and then building spreadsheets to calculate the hedges for those trades as time goes by.
The other is strategy quants. Here you're looking for some way to beat the market. So you're reading in a bunch of data from various sources and applying some sort of quite open-ended analysis on it to come up with rules for a trading system. You then build a framework for trading such systems, which can get very involved eg execution algos that read realtime market data.
I started off in derivatives as a trader but quickly moved on to the other. They're both fairly deep, and both require you to know coding and math.
The questions here seem to apply mostly to derivs quants, but strategy quants have found many things applicable.