<script>
  function filterFunction(param){
  //This method is calling
     jQuery('[id$=myHiddenField]').val(param);
   passStringToController();
  }


  </script>

 <apex:inputHidden value="{!searchBoxstring}" id="myHiddenField"/>
 <apex:actionFunction name="passStringToController" action="{!searchbox}"
 rerender="myHiddenField,msgs">
  <apex:param name="param1" assignTo="{!searchBoxstring}" value="" />
  </apex:actionFunction>
 <input type="text" placeholder="Search.." id="myInput"
   onkeyup="filterFunction('Test');return false;"/>
    <i class="fa fa-search search-icon"></i>
     <button class="close-icon" type="reset"/> </center>
     <apex:pageMessages id="msgs" />

public void searchbox(){
  system.debug('not calling @@@');
 }