Is it possible to call an apex method in javascript quote calculator plugin

One way to do it is wrapping the controller method using an actionFunction, which can then be invoked from JavaScript.

<apex:form>
....    
<apex:actionFunction name="logSearchTermAF" action="{!logSearchTerm}" />
</apex:form>

function searchTerm(){
....
logSearchTermAF();
}

JavaScript Remoting is the other alternative.

jQuery is JavaScript. Just call the function normally. You can call an actionFunction, RemoteAction, and so on.

As a trivial example, here's a simple incremental controller. Every time you call increment, the value increases by 1:

public class counterController {
    public Integer counter { get; set; }
    public counterController() {
        counter = 0;
    }
    public void increment() {
        counter++;
    }
}

In the page, I'll use jQuery to bind to the keypress event, which in turn will call the controller's "increment" function:

<apex:page controller="counterController">
    <script 
    src="https://code.jquery.com/jquery-3.1.0.slim.min.js"  
    integrity="sha256-cRpWjoSOw5KcyIOaZNo4i6fZ9tKPhYYb6i5T9RSVJG8=" 
    crossorigin="anonymous">
        </script>
    <script>
    $(function() {
        $(window).on("keypress", function() {
            alert("I'm about to increment the counter.");
            incrementCounter();
        });
    });
    </script>
    <apex:form id="form">
        Press a key to increment the counter.
        <apex:actionFunction name="incrementCounter" action="{!increment}" reRender="form" />
        The counter is now {!counter}.
    </apex:form>
</apex:page>