SELECT id, name FROM table WHERE name LIKE a%

To change this behaviour, you will need to change sopt parameter in searchoptions in your colModel.
Lets suppose that we want to change the default LIKE a% operator (begins with) with "contains" - i.e LIKE %a% on the field "CustomerID".

Sample code:

<?php
require_once 'jq-config.php';
// include the jqGrid Class
require_once "php/jqGrid.php";
// include the driver class
require_once "php/jqGridPdo.php";
// Connection to the server
$conn = new PDO(DB_DSN,DB_USER,DB_PASSWORD);
// Tell the db that we use utf-8
$conn->query("SET NAMES utf8");

// Create the jqGrid instance
$grid = new jqGridRender($conn);
// Write the SQL Query
$grid->SelectCommand = 'SELECT OrderID, OrderDate, CustomerID, ShipName, Freight FROM orders';
// Set output format to json
$grid->dataType = 'json';
// Let the grid create the model
$grid->setColModel();
// Set the url from where we obtain the data
$grid->setUrl('grid.php');
// Set some grid options
$grid->setGridOptions(array(
    "rowNum"=>10,
    "rowList"=>array(10,20,30),
    "sortname"=>"OrderID"
));

//enable toolbarsearch
$grid->toolbarfilter = true;

//CoustomerID field should be serched with contain operator
$grid->setColProperty('CustomerID',array("searchoptions"=>array("sopt"=>array("cn"))));


$grid->renderGrid('#grid','#pager',true, null, null, true,true);
$conn = null;
?>