How do I implement a nonorthogonal quadratic spline wavelet into discrete wavelet transform?
I would think it is available in the Matlab codes from the Wavelab toolbox. It hosts codes for papers and books. Especially, there is a directory \Wavelab\Books\WaveTour\ for Stéphane Mallat's book: A Wavelet Tour of Signal Processing. You can check images chapter by chapter from 
A Wavelet Tour/Images. Those images can be regenerared. For instance, from Chapter 5,  Wavelab codes in \Matlab_Wavelab\Books\WaveTour\WTCh05 say from instance in wt05fig04.m:


  disp('The dyadic wavelet is a quadratic spline with one vanishing
  moment.') disp('These graphs display the modulus square of its Fourier
  transform') disp('dilated by 2^j for 0


which may suit your needs.