I dont understand what you are trying to say. If you want only the search text box and no buttons, you can use the "onkeyup" or something else in the <apex:inputText> tag. For eg. :

VF Page:

<apex:page controller="contactActivityHistory3">
<apex:form >
      <apex:pageBlock title="Filter">
    <!--      <apex:pageBlockButtons >
              <apex:commandButton value="Fetch" action="{!fetch}" reRender="pagination"/>
          </apex:pageBlockButtons> -->
          <apex:pageBlockSection >
              <apex:inputText value="{!status}" label="Status" onkeyup="{!Fetch}"/>
          </apex:pageBlockSection>
      </apex:pageBlock>
      <apex:pageBlock title="Activity History" id="pagination">
      <apex:pageBlockSection >
          <apex:pageBlockTable value="{!ListTasks}" var="tsk">
              <apex:column headerValue="Contact" value="{!tsk.WhoId}"/>
              <apex:column headerValue="Subject" value="{!tsk.Subject}" />
              <apex:column headerValue="Status" value="{!tsk.Status}"/>
              <apex:column headerValue="Modified Date" value="{!tsk.LastModifiedDate}"/>
          </apex:pageBlockTable>
      </apex:pageBlockSection>  
  </apex:pageBlock>
</apex:form>
</apex:page>

Controler:

public class contactActivityHistory3 {
    
    public Boolean fetchCalled{get;set;}
    public String taskList{get;set;}
    public String status{get;set;}
    public String subject{get;set;}
    
    public contactActivityHistory3() {
        fetchCalled = false;
    }
    
    public void getFetch() {
        taskList = 'SELECT WHOID, Subject, Status, LastModifiedDate FROM Task WHERE Status LIKE \'%'+status+'%\'';
        List<Task> tskList = Database.query(taskList);
        fetchCalled = true;
    }

    public Task[] getListTasks() {
                        List<Contact> cntct = [SELECT Id, Name FROM Contact];
                List<Id> contactIds = new List<Id>();
                for(Contact cn : cntct) {
                contactIds.add(cn.Id);
                }
                
        if (fetchCalled) {
                getfetch();
        }
        
        else {
                        taskList = 'select whoid,subject,status,LastModifiedDate from task where whoid in : contactIds order by LastModifiedDate desc';    
        }
        List<Task> tskList = Database.query(taskList);
                return tskList;            
    }
}

This will have only textbox for Status field (or any other field based on which you need to query the database! You have to change the query accordingly) and once you enter text and hit return ("Enter" key on Windows machine keyboards), it will query and return teh results.