You are likely recalculating the same cell multiple times. For example, let's say you have a 3x3 grid where all cells are 1. You will calculate count(a, 1, 1) twice: once as part of count(a, 0, 1), and once as part of count(a, 1, 0).

You should instead keep track of those values you have calculated so far, and return those where available. For example:

static int count(int a[][], int i, int j) {
    return count(a, i, j, new int[a.length][a[0].length]);
}

static int count(int a[][], int i, int j, int[][] results) {
    if (results[i][j] != 0) {
        return results[i][j];
    }
    int rows = a.length;
    int cols = a[0].length;
    int ret;
    if(a[i][j] == 0)
        ret = 0;
    else if (i == rows - 1 && j == cols - 1)
        ret = a[i][j];
    else if (i == rows - 1)
        ret = a[i][j + 1];
    else if (j == cols - 1)
        ret = a[i + 1][j];
    else if (a[i][j] == 1)
        ret = count(a, i + 1, j, results) + count(a, i, j + 1, results);
    else
        ret = 0;
    result[i][j] = ret;
    return ret;
}

You are likely recalculating the same cell multiple times. For example, let's say you have a 3x3 grid where all cells are 1. You will calculate count(a, 1, 1) twice: once as part of count(a, 0, 1), and once as part of count(a, 1, 0).

You should instead keep track of those values you have calculated so far, and return those where available. For example:

static int count(int a[][], int i, int j) {
    return count(a, i, j, new int[a.length][a[0].length]);
}

static int count(int a[][], int i, int j, int[][] results) {
    if (results[i][j] != 0) {
        return results[i][j];
    }
    int rows = a.length;
    int cols = a[0].length;
    int ret;
    if(a[i][j] == 0)
        ret = 0;
    else if (i == rows - 1 && j == cols - 1)
        ret = a[i][j];
    else if (i == rows - 1)
        ret = a[i][j + 1];
    else if (j == cols - 1)
        ret = a[i + 1][j];
    else if (a[i][j] == 1)
        ret = count(a, i + 1, j, results) + count(a, i, j + 1, results);
    else
        ret = 0;
    results[i][j] = ret;
    return ret;
}