Write a trigger on Lead to prevent conversion if certain custom fields or a related checklist task is incomplete.

trigger LeadTrigger on Lead (before update) {
    Set<Id> leadIds = new Set<Id>();
    for (Lead l : Trigger.new) if (l.IsConverted && !Trigger.oldMap.get(l.Id).IsConverted) leadIds.add(l.Id);
    Map<Id, Integer> openTasks = new Map<Id, Integer>();
    for (AggregateResult ar : [SELECT WhoId w, COUNT(Id) c FROM Task WHERE WhoId IN :leadIds
                               AND IsClosed = false GROUP BY WhoId]) openTasks.put((Id) ar.get('w'), (Integer) ar.get('c'));
    for (Lead l : Trigger.new) {
        if (leadIds.contains(l.Id) && (l.Qualification_Notes__c == null || openTasks.containsKey(l.Id))) {
            l.addError('Complete the required field and close checklist tasks before converting.');
        }
    }
}