public integer monthTest;

then
change this line

Integer m = Date.Today().Month();

to
 

Integer m = Test.isRunningTest() ? monthTest : Date.Today().Month();

then in test class 
Add one line

CI1.monthTest = 5;

Final Code will look something like below
 

global class ScheduleInvoice implements Schedulable {
    public static String CRON_EXP = '0 0 0 1 * ? *';
	public integer monthTest;
    global static String scheduleIt() {
        ScheduleInvoice sm = new ScheduleInvoice();
        return System.schedule('Monthly Reconciliation', CRON_EXP, sm);
    }
    global void execute(SchedulableContext sc) {Id AccountEIDrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('EID').getRecordTypeId();
        Id AccountPVCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('PVC').getRecordTypeId();
        Id AccountPMCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('PMC').getRecordTypeId();
        Id AccountVMCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('VMC').getRecordTypeId();
        Id InvoiceVRCrecordTypeId = Schema.getGlobalDescribe().get('Invoice__c').getDescribe().getRecordTypeInfosByName().get('VRC').getRecordTypeId();
        
        set<ID> recordIds = new Set<ID>();
        recordIds.add(AccountEIDrecordTypeId);
        recordIds.add(AccountPVCrecordTypeId);
        recordIds.add(AccountPMCrecordTypeId);
        recordIds.add(AccountVMCrecordTypeId);
        
        List<Invoice__c> invoiceList = new List<Invoice__c>();
        List<Invoice__c> invList = new List<Invoice__c>();
        Integer m = Test.isRunningTest() ? monthTest : Date.Today().Month();
        Date firstDayOfMonth = System.today().toStartOfMonth();
        Date lastDayOfMonth = firstDayOfMonth.addDays(Date.daysInMonth(firstDayOfMonth.year(), firstDayOfMonth.month()) - 1);
        Date invDesDate = system.Today().addMonths(1);
        Integer invMonth = invDesDate.Month();
        Integer invYear = invDesDate.year();
        String MonthName;
      
        for (Account acct: [
            select Id, Active__c RecordTypeId
            from Account
            where RecordTypeId IN: recordIds and Active__c = true
        ]) {
                Invoice__c inv = new Invoice__c();
                inv.Account__c = acct.Id;
                inv.Status__c = 'Accepted';
                inv.Date__c = system.Today();
                switch on m {
                    when 1 {        
                        invoiceDate__c = 'Jan 1 to Jan 31';
                    }    
                    when 2 {        
                        invoiceDate__c = 'Feb 1 to Feb 28';
                    }
                    when 3 {
                        invoiceDate__c = 'Mar 1 to Mar 31';
                    }
                    when 4 {          
                        invoiceDate__c = 'Apr 1 to Apr 30';
                    }
                     when 5 {        
                        invoiceDate__c = 'May 1 to May 31';
                    }    
                    when 6 {        
                        invoiceDate__c = 'Jun 1 to Jun 30';
                    }
                    when 7 {
                        invoiceDate__c = 'Jul 1 to Jul 31';
                    }
                    when 8 {          
                        invoiceDate__c = 'Aug 1 to Aug 31';
                    }
                    when 9 {        
                        invoiceDate__c = 'Sep 1 to Sep 30';
                    }    
                    when 10 {        
                        invoiceDate__c = 'Oct 1 to Oct 31';
                    }
                    when 11 {
                        invoiceDate__c = 'Nov 1 to Nov 30';
                    }
                    when 12 {          
                        invoiceDate__c = 'Dec 1 to Dec 31';
                    }
                }                                                                                                                
                   inv.Payment_Due_Date__c =  lastDayOfMonth; 
                switch on invMonth {
                    when 1 {        
                        MonthName = 'January';
                    }    
                    when 2 {        
                        MonthName = 'February';
                    }
                    when 3 {
                        MonthName = 'March';
                    }
                    when 4 {          
                        MonthName = 'April';
                    }
                     when 5 {        
                        MonthName = 'May';
                    }    
                    when 6 {        
                        MonthName = 'June';
                    }
                    when 7 {
                        MonthName = 'July';
                    }
                    when 8 {          
                        MonthName = 'August';
                    }
                    when 9 {        
                        MonthName = 'September';
                    }    
                    when 10 {        
                        MonthName = 'October';
                    }
                    when 11 {
                        MonthName = 'November';
                    }
                    when 12 {          
                        MonthName = 'December';
                    }
                }
            inv.RecordTypeId = InvoiceVRCrecordTypeId;
                inv.Description__c = MonthName + ' ' + String.valueof(invYear) + ' ' + 'TMC';
                
               invList.add(inv);
        }
          
        if (!invList.isEmpty()) {
            insert invList; 
        }   
    }
}

===================================
TestClass
 

@isTest
public class CreatingInvoice_Test {
    @isTest static void executeTest(){
        
        Test.startTest();
        
        Id AccountPVCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('PVC').getRecordTypeId();
        Id AccountPMCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('PMC').getRecordTypeId();
        Id AccountVMCrecordTypeId = Schema.getGlobalDescribe().get('Account').getDescribe().getRecordTypeInfosByName().get('VMC').getRecordTypeId();
        Id InvoiceVRCrecordTypeId = Schema.getGlobalDescribe().get('Invoice__c').getDescribe().getRecordTypeInfosByName().get('VRC').getRecordTypeId();
        
        SchedulableContext sc = null;
        ScheduleInvoice CI = new ScheduleInvoice();
        CI.execute(sc);
        
        account a = new account();
        a.Name = 'Test Account';
        a.Active__c = true;
        a.RecordTypeId = AccountVMCrecordTypeId;
        
        insert a;
        
        ScheduleInvoice CI1 = new ScheduleInvoice();
        String sch1 = '0 0 0 1 * ? *'; 
		CI1.monthTest = 1;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 2;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 3;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 4;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 5;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 6;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 7;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 8;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 9;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 10;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 11;
        system.schedule('ScheduleInvoice', sch1, CI1);
		
		CI1.monthTest = 12;
        system.schedule('ScheduleInvoice', sch1, CI1);			
        
        Test.stopTest();
    }
}

Try with this one , let me know how it goes!