Hii Shraddha Ahire,

Yes You can, for this work around you can follow the below Steps

Steps :
Step 1: First we need to use required="false" for that field as shown in the below piece of code.
<apex:inputField value="{!acc.name}" required="false"/>
Step 2: To make that field required use the sample code below.
<apex:pageBlockSectionItem > Account Name
   <apex:outputPanel layout="block" style="float: left">
  <apex:outputPanel >
<div class="requiredInput"><div class="requiredBlock"/>
  <apex:inputField value="{!acc.name}" required="false" />          
</div>
  </apex:outputPanel>                              
   </apex:outputPanel>
</apex:pageBlockSectionItem>

Example :
Apex Class : StandardErrorOverrideController
public class StandardErrorOverrideController {
    public account acc {get;set;}
    public StandardErrorOverrideController() {
        acc = new Account();
    }
    public void saveData() {

        try {
        insert acc;
        } catch(DMLException e) {
            if(acc.name =='' || acc.name == null) {
                acc.name.addError('Account Name Cannot be Null');
            }
        }       
   }    
}
Visualforce Page :
<apex:page controller="StandardErrorOverrideController">
<apex:form >
    <apex:pageBlock title="My Content" mode="edit">
        <apex:pageBlockButtons >
            <apex:commandButton action="{!saveData}" value="Save"/>
        </apex:pageBlockButtons>
        <apex:pageBlockSection title="My Content Section" columns="2">
             <apex:pageBlockSectionItem > Account Name
               <apex:outputPanel layout="block" style="float: left">
                      <apex:outputPanel >
                            <div class="requiredInput"><div class="requiredBlock"/>
                                  <apex:inputField value="{!acc.name}" required="false" />          
                            </div>
                      </apex:outputPanel>                              
               </apex:outputPanel>
            </apex:pageBlockSectionItem>
            <apex:inputField value="{!acc.site}"/>
            <apex:inputField value="{!acc.type}"/>
        </apex:pageBlockSection>
    </apex:pageBlock>
</apex:form>
</apex:page>

Regards,
Ahmed Ansari