How to store a value on WP_option table

replace your code with as.

 <?php
function myplugin_register_settings() 
{
    add_option('marker_pin','');
    register_setting( 'marker_group', 'marker_pin', 'myplugin_callback' );
}
add_action( 'admin_init', 'myplugin_register_settings' );
function myplugin_register_options_page() 
{
    add_options_page('Google Map', 'Map', 'manage_options', 'Google_map', 'myplugin_options_page');
}
add_action('admin_menu', 'myplugin_register_options_page');
function plugin_admin_enqueue_styles() 
{
    wp_enqueue_script( 'google-maps', 'https://maps.googleapis.com/maps/api/js?key=AIzaSyBPUcMLet39Rn_MXvOJErCbT7VnsOAVPWc&libraries=places&callback=initMap', array('jquery'),'1.0', true );
    wp_enqueue_script( 'ajax-js', 'https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js', array('jquery'),'1.0', true );
    wp_enqueue_script( 'code-js', 'https://code.jquery.com/jquery-1.11.3.min.js', array('jquery'),'1.0', true );
}
add_action( 'admin_enqueue_scripts', 'plugin_admin_enqueue_styles' );
function myplugin_options_page()
{?>
    <div>
        <?php screen_icon(); ?>
        <h2>My Plugin Page Title</h2>
        <form method="post" action="options.php">
            <?php settings_fields( 'marker_group' ); ?>
            <h3>This is my option</h3>
            <p>Some text here.</p>
            <table>
                <tr valign="top">
                    <th scope="row"><label for="marker">City:</label></th>
                    <td>
                        <?php  $pins = get_option('marker_pin'); ?>
                        <div id="pac-container">
                            <input id="pac_input" type="text" name="pac_input" placeholder="Enter City" value="<?php echo $pins; ?>">
                        </div>
                    </td>
                </tr>
            </table>
            <?php submit_button(); ?>
            <input type="button" value="Add Button" id='addButton'>
        </form>
    </div><?php
}
function update_option_fn()
{
    if (isset($_POST['name_palce']) && !empty($_POST['name_palce'])) 
    {
       $name=$_POST['name_palce'];
       echo $name;
       update_option('marker_pin',$name);       
    }
}
add_action('wp_ajax_update_option_fn', 'update_option_fn');
?>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
 <script>
        var place="";
        function initMap() 
        {
            var input = document.getElementById("pac_input");
            var autocomplete = new google.maps.places.Autocomplete(input);
            autocomplete.addListener('place_changed', function() 
            {
                var place = autocomplete.getPlace();
                var placename = place.name;
                input.value=placename;
                if (!place.geometry)
                {
                    alert("No details available for input: '" + place.name + "'");
                    return;
                }
            });
        }
        $(document).ready(function() 
        {
            <?php $ajax_nonce = wp_create_nonce( "my-special-string" );?>
            $('#pac_input').focusout(function() 
            {
                var Place =  $('#pac_input').val(); 
                console.log(Place);

                var wp_sec_data_post = { action: 'update_option_fn', security: '<?php echo $ajax_nonce; ?>'};
                var ajaxurl = "<?php echo admin_url('admin-ajax.php')?>";
                wp_sec_data_post2['name_palce'] = Place;
                jQuery.post( ajaxurl, wp_sec_data_post2, function( response ) 
                {
                    console.log("success"+response);                    
                });                
            });
            $("#submit").click(function() 
            {
                var Place =  $('#pac_input').val(); 
                console.log(Place);
                var wp_sec_data_post = { action: 'update_option_fn', security: '<?php echo $ajax_nonce; ?>'};
                var ajaxurl = "<?php echo admin_url('admin-ajax.php')?>";
                wp_sec_data_post2['name_palce'] = Place;
                jQuery.post( ajaxurl, wp_sec_data_post2, function( response ) 
                {
                    console.log("success"+response);                    
                });
            });
        });
</script>