[SalesForce] How To Call A Java Class Method from Salesforce

With out any Visualforce Page this can be done using Ajax Toolkit.

Below example Shows updating one of the field on click of Button..Create a button -> Execute Javascript & on Click Javascript.

{!requireScript("/soap/ajax/23.0/connection.js")}

 var newTask= new sforce.SObject("Task");

 newTask.id = "{!Task.Id}";


 newTask.Read_Status__c="Read";


 var result = sforce.connection.update([newTask]);

 if (result[0].getBoolean("success"))
 {
   // Refresh window
    window.location.reload();
 }
 else
{
   alert("something broke");
 }

This question is very similar to the one you already asked here: how to create a custom button for a custom object on a related list

If I don't misunderstand you requirements, you got a lot of things conceptually wrong.

First Competitors are related to quotes and you want to find all competitors for that quote. So you have to bypass the QuoteId as filter criteria - not the CompetitorId.

So your button code could be something like

/apex/CompitetorsForQuote?id={!Quote.Id}

There is also absolutely no need for using Javascript as Button type. You should use URL.

Then you query for a single competitor (using the wrong where) but you should query for a list.

string QuoteId = System.currentPageReference().getParameters().get('Id');
Competitor__c[] comps = [SELECT Id,Name,Product_Series__c,   Part_Number__c,  
    Price_Offered__c , Volume__c,  Date_Price_is_valid__c FROM Compitetor__c 
    WHERE Quote__c = :QuoteId;
];

This is probably not all and my code is untested and may contain errors. But it could help you to think about the structure.