A starting point, as I have no time to finish this, even less to test this.
Most of the work is to remove Java fluff...
You will have to manage the paint() so it doesn't get a Graphics, but instead paints on the sketch itself.
You better load the image using Processing idioms directly.
You have to guess what the two Thread classes, whose code is missing here, are doing. Perhaps they don't need to be Threads, anyway.
I hope to give you a general idea of the work to do.

MatrixDisplay matrix;

void setup()
{
  size(400, 250);

  matrix = new MatrixDisplay();
}

void draw()
{
  matrix.paint();
}

public class MatrixDisplay {
  public int vRows[]; //unused
  public int vCols[]; //unused
  public int nactive[]; //number of active cells in the column ('stream length')
  public int offsets[]; //offsets of the columns from 0
  public int velos[]; //velocities of the columns (right now, 0-4)
  public int probs[]; //velocities of the columns (right now, 0-4)
  public int upd[];
  public int nRows = 0; //number of rows (estimated)
  public int nCols = 0; //number of columns (estimated)

  public int vI;

  public char[] carr;
  public char[][] matrix;

  // Classes are missing, you have to guess what they do and skip the Thread part, perhaps
  private RCGthread RCG;
  private RNGthread RNG;

  public Color flash;
  public Color tail;
  public Color ngreen;

  //constructor
  public MatrixDisplay() {
    nCols = width/15;
    nRows = height/15;
    try {
      matrix = new char[nCols][nRows];
      nactive = new int[nCols];
      offsets = new int[nCols];
      velos = new int[nCols];
      probs = new int[nCols];
      upd = new int[nCols];

      //setup initial r/c/v s
      for (int i = 0; i < nCols; i++)
      {
        nactive[i] = (int) random(nRows/1.5);
        offsets[i] = 15-(nactive[i]*15);
        velos[i] = (int) random(12)+2;
        probs[i] = (int) random(10);
        upd[i] = 0;
      }
    }
    catch(Exception ex) {
      println("Initialization error:"+ ex);
      println("Try resizing your window to fix it");
    }
    flash = new Color(230, 255, 230);
    tail = Color.green.darker().darker();
    ngreen = Color.green.darker();

    RCG = new RCGthread(10240);
    RCG.start();
    RNG = new RNGthread(10240);
    RNG.start();

    carr = new char[1];
  }

  public void update (Graphics g) {
    if ( backImage == null || backImage.getWidth(this) != this.getSize().width ||
      backImage.getHeight(this) != this.getSize().height ) {
      MediaTracker mt = new MediaTracker( this );
      backImage = createImage( this.getSize().width, this.getSize().height );
      mt.addImage( backImage, 0 );
      try {
        mt.waitForAll();
      }
      catch(Exception e) {
      }
      bg = backImage.getGraphics();
    }
    if ( bg != null && this != null ) {
      bg.setColor( Color.black );
      bg.setFont(new Font("serif", Font.BOLD, 15));
      bg.fillRect( 0, 0, this.getSize().width, this.getSize().height );
    }
    paint(bg);
    if (backImage != null) g.drawImage(backImage, 0, 0, this);
  }

  //double buffered painting...
  public void paint(Graphics g)
  {
    int colctr;
    try {
      for (int il = 0; il < nCols; il++)
      {
        double tempR = RNG.get_RNG();

        if (offsets[il] < nactive[il]*15)
        {
          offsets[il] += 15;
          upd[il] = 3;
        }
        else if ((int)(tempR*96) <= ( probs[il] * (velos[il]-5)))
        {
          offsets[il] += 15 ;
          upd[il] = 2;
        }

        else if ((int)(RNG.get_RNG()*100) <= ( probs[il] * (velos[il]-5)))
        {
          offsets[il] = offsets[il] + velos[il] ;
          upd[il] = 1;
        }

        else
        {
          upd[il] = 0;
        }
      }
    }
    catch(Exception ex) {/*System.err.println("paint caught:"+ex);*/
    }
    int h = height;
    int jptemp;
    int iptemp;
    try {
      //loop, updating each column's position
      for (int i = 1; i < nCols; i++ )
      {
        iptemp = i * 15;
        colctr = nactive[i];

        for (int j = 0; j <= colctr; j++ )
        {
          jptemp = j * 15;

          if ((j == 0 ) /*&& (upd[i] == 1)*/)
          {
            bg.setColor(tail);
          }
          else if (j == 2)
          {
            bg.setColor(ngreen);
          }

          if (upd[i] > 1)   
          {
            if (j == colctr)
            {
              if (upd[i] == 2)
                bg.setColor(flash);
              matrix[i][j] = RCG.get_RCG();
            }
            else
              matrix[i][j] = matrix[i][j+1];

            carr[0] = matrix[i][j];
            bg.drawChars( carr,
            0, 1, (iptemp), ( (offsets[i]) + (jptemp) ) % h );
          }

          else if (upd[i] == 1 )
          {

            if (j == colctr)
              bg.setColor(flash);

            carr[0] = RCG.get_RCG();
            matrix[i][j] = carr[0];
            bg.drawChars( carr,
            0, 1, (iptemp), ( (offsets[i]) + (jptemp) ) % h );
          }
          else
          {
            if (matrix[i][j] == '\0')
            {
              carr[0] = RCG.get_RCG();
              matrix[i][j] = carr[0];
            }
            else
              carr[0] = matrix[i][j];
            bg.drawChars( carr,
            0, 1, (iptemp),
            ( (offsets[i] ) + (jptemp) ) % h );
          }
        }
      }
    }
    catch(Exception ex) {/*System.err.println("paint error:"+ex);*/
    }
  }
}