Hitesh,

Now I have modified class as follows :

global class scheduledUpdateOnOppurtunity implements Database.Batchable<SObject>, Database.AllowsCallouts, Schedulable{
 
public String query ;
public Date currentDate =  date.today();
public Integer monthDays =  date.daysInMonth(currentDate.year(), currentDate.month())-1;
public Date startOfMonth = currentDate.toStartOfMonth();
public Date endOfMonth = startOfMonth.addDays(monthDays);         
public Integer currentFinancialYear = currentDate.year();
public date currentQuarterStart;
public date currentQuarterEnd;
public date firstQuarterStart;
public date firstQuarterEnd;
public date secondQuarterStart;
public date secondQuarterEnd;
public date thirdQuarterStart;
public date thirdQuarterEnd;
public date forthQuarterStart;
public date forthQuarterEnd;

global database.Querylocator start(Database.BatchableContext BC){
       query = 'SELECT CloseDate, Closed_date_Today__c,Closed_date_StartOfMonth__c, Closed_date_EndOfMonth__c, Closed_date_CurrentQuarterStart__c, Closed_date_CurrentQuarter__c from Opportunity WHERE isClosed = false';
       return Database.getQueryLocator(query);
}

global void execute(Database.BatchableContext BC,Sobject[] scope)
{
          if(currentDate.month() <=3 ){
              currentFinancialYear = currentDate.year() - 1;
          }
          firstQuarterStart = date.newInstance(currentFinancialYear, 04, 01);
          firstQuarterEnd = date.newInstance(currentFinancialYear, 06, 30);
          
          secondQuarterStart = date.newInstance(currentFinancialYear, 07, 01);
          secondQuarterEnd = date.newInstance(currentFinancialYear, 09, 30);
          
          thirdQuarterStart = date.newInstance(currentFinancialYear, 10, 01);
          thirdQuarterEnd = date.newInstance(currentFinancialYear, 12, 31);
          
          forthQuarterStart = date.newInstance(currentFinancialYear+1, 01, 01);
          forthQuarterEnd = date.newInstance(currentFinancialYear+1, 03, 31);        
          
        
          if (currentDate >= firstQuarterStart && currentDate <= firstQuarterEnd ){
              currentQuarterStart = firstQuarterStart;
              currentQuarterEnd = firstQuarterEnd;
          }
          
          if (currentDate >= secondQuarterStart && currentDate <= secondQuarterEnd ){
              currentQuarterStart = secondQuarterStart;
              currentQuarterEnd = secondQuarterEnd;
          }
          
          if (currentDate >= thirdQuarterStart && currentDate <= thirdQuarterEnd ){
              currentQuarterStart = thirdQuarterStart;
              currentQuarterEnd = thirdQuarterEnd;
          }
          
          if (currentDate >= forthQuarterStart && currentDate <= forthQuarterEnd ){
              currentQuarterStart = forthQuarterStart;
              currentQuarterEnd = forthQuarterEnd;
          }         
            
           try{
                List<Opportunity> listOppUpdate = new List<Opportunity>();
                           
                 for(sobject s : scope){
                      Opportunity opp = (Opportunity)s;
                
                //if Close Date is less than Today then set Closed_date_Today__c to true else set it to false
                if (opp.CloseDate > currentDate){
                    opp.Closed_date_Today__c = true;
                }else{        
                    opp.Closed_date_Today__c = false;
                }
        
                //if Close Date is greater than equal to start of month then set Closed_date_StartOfMonth__c to true else set it to false
                if (opp.CloseDate >= startOfMonth){
                    opp.Closed_date_StartOfMonth__c = true;
                }else{
                    opp.Closed_date_StartOfMonth__c = false;
                }
        
                //if Close Date is in in the current month and is greater than Today then set Closed_date_EndOfMonth__c to true else set it to false
                if (opp.CloseDate <= endOfMonth){
                    opp.Closed_date_EndOfMonth__c = true;
                }else{
                    opp.Closed_date_EndOfMonth__c = false;
                }
        
                //if Close Date is greater than Start of Current quarter then set Closed_date_CurrentQuarterStart__c to true else set it to false
                if(opp.CloseDate >= currentQuarterStart){
                    opp.Closed_date_CurrentQuarterStart__c = true;
                }else{
                    opp.Closed_date_CurrentQuarterStart__c = false;
                }         
        
                //if Close Date lies in the current quarter then set Closed_date_CurrentQuarter__c to true else set it to false
                if(opp.CloseDate <= currentQuarterEnd){
                    opp.Closed_date_CurrentQuarter__c = true;
                }else{
                    opp.Closed_date_CurrentQuarter__c = false;
                }     
                    
                   listOppUpdate.add(opp);
                    
                   if(listOppUpdate.size() > 999){
                       database.update(listOppUpdate,false);
                        listOppUpdate = new List<Opportunity>();
                   }
                 }
                 
                 if(listOppUpdate.size() > 0){
                     database.update(listOppUpdate,false);
                 }
                 
              }
              catch(Exception ex){
                  system.debug(ex.getMessage());
              }  
      
  }
 
  global void finish(Database.BatchableContext BC){
  }

  global void execute(SchedulableContext context) {
    // call batch apex
    scheduledUpdateOnOppurtunity batchObj = new scheduledUpdateOnOppurtunity();
    Database.executeBatch(batchObj);
  }
 
}

And test method

@isTest
private class TestSchedulableUpdateOpportunity {  
      
    public static testmethod void DoTestSch(){
        
        Test.startTest();
        scheduledUpdateOnOppurtunity batchObj = new scheduledUpdateOnOppurtunity();
        SchedulableContext context;
        batchObj.execute(context);
        Test.stopTest();     
    }
}

But coverage is still 30% :(