Depth first search - 2D Game map

I have created a 2D maze, and I want to find the quickest path between the red->blue colored nodes. I'm an unsure how I would go about implementing a depth-first search. I know that an adjacency matrix or list could be used to represent the connections between nodes. Though, I am unsure of how to construct it.

for brevity: I need to return a list with the tiles coordinates searched (when looking for goal node), so i can depict the search on the maze. Or how would i construct an adjacency matrix for this? and the corresponding vertex list?

Depth first search general structure

1. Visit node(cell) (change visited flag to true)
2. Push to Stack
3. get unvisited vertex (peek stack) if none (pop stack) - update maze model view

Repeat 1 - 3 until stack empty

Here is the current code for the maze class.

public class Maze {

    //Tile ids
    public static short OBSTICLE = 0;
    public static short START_LOC_VALUE = -2;
    public static short GOAL_LOC_VALUE = -3;

    private int rows, cols;
    private int numTiles;
    private int[][] map;
    private int[][] adjMatrix;
    private Queue theQueue; 

    public Maze(int rows, int cols){
        this.rows = rows;
        this.cols = cols;

        theQueue = new Queue();

        numTiles = rows*cols;

        map = new int[rows][cols];
        adjMatrix = new int[rows][cols];

        for (int i=0; i<rows; i++) {
            for (int j=0; j<cols; j++) {
                map[i][j] = 1;
            }
        }
    }

    /*
     * Generate Maze
     * @param numObstacles - number of obstacles
     */
    public void generateMaze(int numObstacles){
        for (int i = 0; i < numObstacles; i++)
            setTile((int)(Math.random()*rows),(int)(Math.random()*cols), Maze.OBSTICLE);

       //setTile((int)(Math.random()*rows),(int)(Math.random()*cols),Maze.START_LOC_VALUE);
       //setTile((int)(Math.random()*rows),(in