You could create a custom validation attribute like this:
public class UniqueAnswersOnly : ValidationAttribute
{
    protected override ValidationResult IsValid(object value, ValidationContext validationContext)
    {
        //Get a list of all properties that are marked with [UniqueAnswersOnly]
        var props = validationContext.ObjectInstance.GetType().GetProperties().Where(
            prop => Attribute.IsDefined(prop, typeof(UniqueAnswersOnly)));
        var values = new HashSet<string>();
        //Read the values of all other properties
        foreach(var prop in props)
        {
            var pValue = (string)prop.GetValue(validationContext.ObjectInstance);
            if (prop.Name!=validationContext.MemberName && !values.Contains(pValue))
            {
                values.Add(pValue);
            }
        }
        if (values.Contains(value))
        {
            return new ValidationResult("Duplicate answer", new[] { validationContext.MemberName });
        }
        return null;
    }
}
and here is a test case:
public class SecurityQuestions
{
    public int Question1Id { get; set; }
    public int Question2Id { get; set; }
    public int Question3Id { get; set; }
    public int Question4Id { get; set; }
    public int Question5Id { get; set; }
    public int Question6Id { get; set; }
    [UniqueAnswersOnly]
    public string Answer1 { get; set; }
    [UniqueAnswersOnly]
    public string Answer2 { get; set; }
    [UniqueAnswersOnly]
    public string Answer3 { get; set; }
    [UniqueAnswersOnly]
    public string Answer4 { get; set; }
    [UniqueAnswersOnly]
    public string Answer5 { get; set; }
    [UniqueAnswersOnly]
    public string Answer6 { get; set; }
}
class Program
{
    static void Main(string[] args)
    {
        var questions = new SecurityQuestions();
        questions.Answer1 = "Test";
        questions.Answer2 = "Test";
        questions.Answer3 = "Test3";
        questions.Answer4 = "Test4";
        questions.Answer5 = "Test5";
        questions.Answer6 = "Test6";
        var vc = new ValidationContext(questions, null, null);
        var results = new List<ValidationResult>();
        var validationResult = Validator.TryValidateObject(questions, vc, results, true);
    }
}
Edit:
I created a default MVC project and added your current code. That validation works just fine.
I added this to the home controller:
    public ActionResult AskQuestions()
    {
        var questions = new SecurityQuestions();
        return View(questions);
    }
    [HttpPost]
    public ActionResult CheckQuestions(SecurityQuestions questions)
    {
        if (ModelState.IsValid)
        {
            return View();
        }            
        else
        {
            return HttpNotFound();
        }
    }
And the AskQuestions view looks like this:
@model WebApplicationValidation.Models.SecurityQuestions
@{
    ViewBag.Title = "AskQuestions";
}
<h2>AskQuestions</h2>
@using (Html.BeginForm("CheckQuestions", "Home",FormMethod.Post))
{
    @Html.AntiForgeryToken()
    <div class="form-horizontal">
        <h4>SecurityQuestions</h4>
        <hr />
        @Html.ValidationSummary(true, "", new { @class = "text-danger" })
        <div class="form-group">
            @Html.LabelFor(model => model.Question1Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question1Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question1Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Question2Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question2Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question2Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Question3Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question3Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question3Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Question4Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question4Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question4Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Question5Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question5Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question5Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Question6Id, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Question6Id, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Question6Id, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer1, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer1, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer1, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer2, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer2, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer2, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer3, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer3, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer3, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer4, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer4, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer4, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer5, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer5, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer5, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            @Html.LabelFor(model => model.Answer6, htmlAttributes: new { @class = "control-label col-md-2" })
            <div class="col-md-10">
                @Html.EditorFor(model => model.Answer6, new { htmlAttributes = new { @class = "form-control" } })
                @Html.ValidationMessageFor(model => model.Answer6, "", new { @class = "text-danger" })
            </div>
        </div>
        <div class="form-group">
            <div class="col-md-offset-2 col-md-10">
                <input type="submit" value="Create" class="btn btn-default" />
            </div>
        </div>
    </div>
}
<div>
    @Html.ActionLink("Back to List", "Index")
</div>
@section Scripts {
    @Scripts.Render("~/bundles/jqueryval")
}
If I run the app the ModelState.IsValid() is false if I enter two identical answers. And setting a breakpoint in the validation routine shows that it's being run.