You can set the value before or after initialization. If doing so after, you must use refresh method.
$(function() {
$("#Mon").selectmenu();
$("#set-mon").click(function() {
$("#Mon").val("06").selectmenu("refresh");
});
});<link rel="stylesheet" href="//code.jquery.com/ui/1.12.1/themes/base/jquery-ui.css">
<script src="https://code.jquery.com/jquery-1.12.4.js"></script>
<script src="https://code.jquery.com/ui/1.12.1/jquery-ui.js"></script>
<select class="date1" id="Mon" name="Mon">
<option value="01" id="Mon01">01</option>
<option value="02" id="Mon02">02</option>
<option value="03" id="Mon03">03</option>
<option value="04" id="Mon04">04</option>
<option value="05" id="Mon05">05</option>
<option value="06" id="Mon06">06</option>
<option value="07" id="Mon07">07</option>
<option value="08" id="Mon08">08</option>
<option value="09" id="Mon09">09</option>
<option value="10" id="Mon10">10</option>
<option value="11" id="Mon11">11</option>
<option value="12" id="Mon12">12</option>
</select>
<button id="set-mon">Set Month</button>