Hi jmott,

Please refer below sample.

HTML

<asp:TextBox ID="txtName" runat="server" />
<br />
<div id="dvCaptcha">
</div>
<asp:TextBox ID="txtCaptcha" runat="server" Style="display: none" />
<asp:RequiredFieldValidator ID="rfvCaptcha" ErrorMessage="Captcha validation is required." ControlToValidate="txtCaptcha"
    runat="server" ForeColor="Red" Display="Dynamic" />
<br />
<br />
<asp:Button ID="btnSubmit" Text="Submit" runat="server" OnClick="Submit" />
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript" src="https://www.google.com/recaptcha/api.js?onload=onloadCallback&render=explicit"
    asyncdefer></script>
<script type="text/javascript">
    var onloadCallback = function () {
        grecaptcha.render('dvCaptcha', {
            'sitekey': '<%=ReCaptcha_Key %>',
            'callback': function (response) {
                $.ajax({
                    type: "POST",
                    url: "Default.aspx/VerifyCaptcha",
                    data: "{response: '" + response + "'}",
                    contentType: "application/json; charset=utf-8",
                    dataType: "json",
                    success: function (r) {
                        var captchaResponse = jQuery.parseJSON(r.d);
                        if (captchaResponse.success) {
                            $("[id*=txtCaptcha]").val(captchaResponse.success);
                            $("[id*=rfvCaptcha]").hide();
                        } else {
                            $("[id*=txtCaptcha]").val("");
                            $("[id*=rfvCaptcha]").show();
                            var error = captchaResponse["error-codes"][0];
                            $("[id*=rfvCaptcha]").html("RECaptcha error. " + error);
                        }
                    }
                });
            }
        });
    };
    $(function () {
        $("[id*=txtCaptcha]").val("");
    });
</script>

Namespaces

C#

using System.Net;
using System.Web.Services;

VB.Net

Imports System.Net
Imports System.Web.Services

Code

C#

protected static string ReCaptcha_Key = "<RECaptcha Site Key>";
protected static string ReCaptcha_Secret = "<RECaptcha Secret Key>";

[WebMethod]
public static string VerifyCaptcha(string response)
{
    string url = "https://www.google.com/recaptcha/api/siteverify?secret=" + ReCaptcha_Secret + "&response=" + response;
    return (new WebClient()).DownloadString(url);
}

protected void Submit(object sender, EventArgs e)
{
    ClientScript.RegisterStartupScript(this.GetType(), "alert", "alert('" + txtName.Text + "');", true);
}

VB.Net

Protected Shared ReCaptcha_Key As String = "<RECaptcha Site Key>"
    Protected Shared ReCaptcha_Secret As String = "<RECaptcha Secret Key>"

    <WebMethod>
    Public Shared Function VerifyCaptcha(ByVal response As String) As String
        Dim url As String = "https://www.google.com/recaptcha/api/siteverify?secret=" & ReCaptcha_Secret & "&response=" & response
        Return (New WebClient()).DownloadString(url)
    End Function

    Protected Sub Submit(ByVal sender As Object, ByVal e As EventArgs)
        ClientScript.RegisterStartupScript(Me.GetType(), "alert", "alert('" & txtName.Text & "');", True)
    End Sub

Screenshot