Are you sure you are passing a valid id; how have you modified the AJAX call, and the designationAjax action?

modified code

 $this->validate($request,[
                                'emp_fname'=>'unique:employees|required',
                                'designation_id' => 'exists:designations,designation_id' // checks if the is a record matching the provided value
                                ]);

controller

public function designationAjax($department)
    {
        $designation = Designation::where("department_id",$department)
                    ->pluck('designations.designation_name','designations.designation_id')->all();
        return json_encode($designation);
    }

script

function getDesignation(department) {
    if(department) {
            $.ajax({
                url: '{{url('/')}}/designation/ajax/'+department,
                type: "GET",
                dataType: "json",
                success:function(data) {

                    console.log(data);

                    $('select[name="designation_id"]').empty();
                    $('select[name="designation_id"]').prepend('<option value="">--Select designation--</option>');
                    $.each(data, function(key, value) {
                        $('select[name="designation_id"]').append('<option value="'+ key +'">'+ value +'</option>');
                    });

                }
            });
        } else{
            $('select[name="designation_id"]').empty();
        }
}

Interesting. So, you are getting the value 2 for designation_id in the request>

when i selected for 2nd option for department_id =2 its gets designation_id=0

using department_id=2 whenever i seleted any option still it getting designation_id=0

@tykus please help !!