How can I attach the image or icon on edge of google line chart

use the getChartLayoutInterface method

var layout = chart.getChartLayoutInterface();

to find the (x, y) location of the point

var xPos = layout.getXLocation(data.getValue(row, xColumn));
var yPos = layout.getYLocation(data.getValue(row, yColumn));

then add the image manually and adjust accordingly

google.charts.load('current', {packages: ['corechart', 'line']});
google.charts.setOnLoadCallback(drawBackgroundColor);

function drawBackgroundColor() {
  var data = new google.visualization.DataTable();
  data.addColumn('number', 'X');
  data.addColumn('number', 'Y');

  data.addRows([
    [0, 0],   [1, 10],  [2, 23],  [3, 17],  [4, 18],  [5, 9],
    [6, 11],  [7, 27],  [8, 33],  [9, 40],  [10, 32], [11, 35],
    [12, 30], [13, 40], [14, 42], [15, 47], [16, 44], [17, 48],
    [18, 52], [19, 54], [20, 42], [21, 55], [22, 56], [23, 57],
    [24, 60], [25, 50], [26, 52], [27, 51], [28, 49], [29, 53],
    [30, 55], [31, 60], [32, 61], [33, 59], [34, 62], [35, 65],
    [36, 62], [37, 58], [38, 55], [39, 61], [40, 64], [41, 65],
    [42, 63], [43, 66], [44, 67], [45, 69], [46, 69], [47, 70],
    [48, 72], [49, 68], [50, 66], [51, 65], [52, 67], [53, 70],
    [54, 71], [55, 72], [56, 73], [57, 75], [58, 70], [59, 68],
    [60, 64], [61, 60], [62, 65], [63, 67], [64, 68], [65, 69],
    [66, 70], [67, 72], [68, 75], [69, 80]
  ]);

  var options = {
    colors: ['#000000'],
    legend: 'none',
    lineSize: 2
  };

  var container = document.getElementById('chart_div');
  var chart = new google.visualization.LineChart(container);

  google.visualization.events.addListener(chart, 'ready', function () {
    var layout = chart.getChartLayoutInterface();
    for (var i = 0; i < data.getNumberOfRows(); i++) {
      // add image above every fifth element
      if ((i % 5) === 0) {
        var xPos = layout.getXLocation(data.getValue(i, 0));
        var yPos = layout.getYLocation(data.getValue(i, 1));

        var whiteHat = container.appendChild(document.createElement('img'));
        whiteHat.src="http://findicons.com/files/icons/512/star_wars/16/clone_old.png";
        whiteHat.className="whiteHat";

        // 16x16 (image size in this example)
        whiteHat.style.top = (yPos - 16) + 'px';
        whiteHat.style.left = (xPos) + 'px';
      }
    }
  });

  chart.draw(data, options);
}

.whiteHat {
  border: none;
  position: absolute;
}

<script src="https://www.gstatic.com/charts/loader.js"></script>
<div id="chart_div"></div>

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