Hi RaginiReddy,

Go through this sample code to implement lead conversion through REST API Call.

You can achieve this by this by creating an Apex class for REST call.

Sample code is below for your reference. 

@RestResource(urlMapping='/Lead/*')
global with sharing class RestLeadConvert {            

    @HttpGet
    global static String doGet() {
        String ret = 'fail';
        RestRequest req = RestContext.request;
        RestResponse res = RestContext.response;
        String leadId = req.requestURI.substring(req.requestURI.lastIndexOf('/')+1);              
        Database.LeadConvert lc = new Database.LeadConvert();
        lc.setLeadId(leadId);

        LeadStatus convertStatus = [SELECT Id, MasterLabel FROM LeadStatus WHERE IsConverted=true LIMIT 1];
        lc.setConvertedStatus(convertStatus.MasterLabel);           
        Database.LeadConvertResult lcr ;
        try{
            lcr = Database.convertLead(lc);
            system.debug('*****lcr.isSuccess()'+lcr.isSuccess());            
            ret = 'ok';
        }
        catch(exception ex){
            system.debug('***NOT CONVERTED**');           
        }
        return ret;
    }   
}

And you can use this call by

<Your Instance URL>/services/apexrest/Lead/<LeadId>

This test will give you around 93% of coverage.

@isTest
public class RestLeadConvertTest{

    static testMethod void testHttpGet() {
        Lead l = new Lead();
        l.FirstName = 'First';
        l.LastName = 'Last';
        insert l;

        Test.startTest();
        RestRequest req = new RestRequest();
        RestResponse res = new RestResponse();
        req.requestURI = '/Lead/' + l.Id;
        req.httpMethod = 'GET';
        RestContext.request = req;
        RestContext.response= res;
        RestLeadConvert.doGet();
        Test.stopTest();
    }

}

Hope this helps.

Kindly mark this as solved if the reply was helpful so that it gets removed from the unanswered queue which results in helping others who are facing a similar issue.

Thanks,
Nagendra