You need to make ShowCount() accessible from outside the constructor function. You can do this by attaching the function to a ShowCount property of the returned instance. You then would need to call mySingleton.ShowCount():
var mySingleton = new function()
{
    this.count = 123;
    document.write("<div onclick='mySingleton.ShowCount();'>Click</div>");
    this.ShowCount = function() {
        alert(this.count);    
    };
}();
 
 
Remark: This is just to demonstrate how to fix your implementation, but as already pointed out in the comments above, it is not good style.
A better implementation would create the element and attach its event handlers using the DOM API while avoiding document.write(). Also, your singleton is actually none, so better rename it to something else to avoid confusion (or implement a real singleton):
function myClass() {
  this.count = 123;
  
  // Create <div> and attach onclick handler:
  let div = document.createElement('div');
  div.textContent = 'Click';
  div.onclick = () => alert(this.count);
  document.body.appendChild(div);
};
let instance = new myClass();
 
 
Depending on your use-case, it might still be unusual to have a constructor function create DOM elements.