Instead of using the Page link you can use javascript for that and any div you can show or hide in it.
Your Code:
MenuItem childItem = new MenuItem();
childItem.NavigateUrl = "OtherPage.aspx";
Change with:
MenuItem childItem = new MenuItem();
childItem.NavigateUrl = "javascript: return GoToSomeLink('"+ count +"');"; //You can pass parameters also
Javascript function:
<script type="text/javascript">
function GoToSomeLink(obj) //if parameters are used use them here also.
{
var count=parseInt(obj); //use this count varible anywhere in the function
$(#menuDiv).show(); /any div show or hide
return false;
}
</script>