Show global Message in User Profiles with admin only Input field in WordPress Backend

You just need to move the capability check “inwards”:

function wpse_230369_quote_of_the_day( $user ) { 
    $quote = esc_attr( get_option( 'quote_of_the_day' ) );
    ?>

<div class="visible-only-for-admin">
    <h3>Quote of the Day Input Field</h3>
    <table class="form-table" >
        <tr>
            <th><label for="quote_of_the_day">Quote of the Day</label></th>
            <td>
                <?php if ( current_user_can( 'administrator' ) ) : ?>

                    <input type="text" name="quote_of_the_day" value="<?php echo $quote ?>" class="regular-text" />

                <?php else : ?>

                    <?php echo $quote ?>

                <?php endif ?>
            </td>
        </tr>
    </table>
</div>

<?php
}

add_action( 'show_user_profile', 'wpse_230369_quote_of_the_day', 10 );
add_action( 'edit_user_profile', 'wpse_230369_quote_of_the_day', 10 );

function wpse_230369_save_quote_of_the_day( $user_id ) {
    if ( isset( $_POST['quote_of_the_day'] ) && current_user_can( 'administrator' ) )
        update_option( 'quote_of_the_day', sanitize_text_field( wp_unslash( $_POST['quote_of_the_day'] ) ) );
}

add_action( 'edit_user_profile_update', 'wpse_230369_save_quote_of_the_day' );
add_action( 'personal_options_update',  'wpse_230369_save_quote_of_the_day' );