I've tried using postman and single and double quotes do not make any difference either
 
I can filter the results but it seems like i must be doing something wrong
 
https://api.smartsheet.com/2.0/search/sheets/798272320235396?query="'Elliott'"
 
returns 
{
  "results": [
    {
      "text": "White cottage",
      "objectType": "row",
      "objectId": 3223433130600324,
      "parentObjectType": "sheet",
      "parentObjectId": 798272320235396,
      "parentObjectName": "Leads",
      "contextData": [
        "Row 24: 2003"
      ]
 
    }, etc etc etc