Created a new module for this.

Step 1:

in app/code/local/Contact/Captcha/etc/config.xml,

<?xml version="1.0"?>
<config>
<modules>
    <Contact_Captcha>
        <version>0.1.0</version>
    </Contact_Captcha>
</modules>
<default>
    <captcha>
        <frontend>
            <areas>
                <contact_page_captcha>
                    <label>Contact Page</label>
                </contact_page_captcha>
            </areas>
        </frontend>
    </captcha>
</default>
 <global>
    <models>
        <contact_captcha>
            <class>Contact_Captcha_Model</class>
        </contact_captcha>
    </models>
    <events>
        <controller_action_predispatch_contacts_index_post>
            <observers>
                <captcha>
                    <type>model</type>
                    <class>contact_captcha/observer</class>
                    <method>checkContactPage</method>
                </captcha>
            </observers>
        </controller_action_predispatch_contacts_index_post>
    </events>
</global>
</config>

Step 2:

in app/etc/modules/Contact_Captcha.xml

<?xml version="1.0"?>
<config>
<modules>
    <Contact_Captcha>
        <active>true</active>
        <codePool>local</codePool>
    </Contact_Captcha>
</modules>
</config>

Step 3:

in app/code/local/Contact/Captcha/Model/Observer.php

<?php

class Contact_Captcha_Model_Observer extends Mage_Captcha_Model_Observer
{
public function checkContactPage($observer)
{
    $formId = 'contact_page_captcha';
    $captchaModel = Mage::helper('captcha')->getCaptcha($formId);
    if ($captchaModel->isRequired()) {
        $controller = $observer->getControllerAction();
        if (!$captchaModel->isCorrect($this->_getCaptchaString($controller->getRequest(), $formId))) {
            Mage::getSingleton('customer/session')->addError(Mage::helper('captcha')->__('Incorrect CAPTCHA.'));
            $controller->setFlag('', Mage_Core_Controller_Varien_Action::FLAG_NO_DISPATCH, true);
            $controller->getResponse()->setRedirect(Mage::getUrl('*/*/'));
        }
    }
    return $this;
}
}

Step 4:

Create a local.xml in theme path.

app/design/frontend/theme/default/layout/local.xml

<?xml version="1.0"?>
<layout version="0.1.0">
 <contacts_index_index translate="label">
    <reference name="contactForm">
        <block type="core/text_list" name="form.additional.info">
            <block type="captcha/captcha" name="captcha">
                <reference name="head">
                    <action method="addJs"><file>mage/captcha.js</file></action>
                </reference>
                <action method="setFormId"><formId>contact_page_captcha</formId></action>
                <action method="setImgWidth"><width>230</width></action>
                <action method="setImgHeight"><width>50</width></action>
            </block>
        </block>
    </reference>
</contacts_index_index>
</layout>

Step 4: in app/design/frontend/theme/default/template/contacts/form.phtml

Add below code:

 <?php echo $this->getChildHtml('form.additional.info'); ?>

Step 5:

Admin Settings:

Dashboard - > Customers - > Customer Configuration - > CAPTCHA - > Forms - > Select Contact Page

Step 6:

Clear cache. Refresh contact us page.