Step 1: Configure the Google reCAPTCHA and get the Site Key & Secret Key

• Login to Google reCAPTCHA
• Register a new site
• Add Label e.g. Salesforce.com
• Select reCAPTCHA type “reCAPTCHA v2”
• Select “I’m not a robot” Checkbox option.
• Add a domain e.g. yourorgurl.com
• Accept the reCAPTCHA Terms of Service
• Submit and get the Site Key & Secret Key

Step 2: Add Google as a Remote Site in Salesforce

Step 3: Create apex controller

/*
@Author : Biswajeet Samal
@CreatedDate : 30th JUL 2020
@Description : Google ReCAPTCHA Controller
*/
public class GoogleReCAPTCHAController {
    
    public Boolean verified { get; private set; }
    public String response  { 
        get {
            return ApexPages.currentPage().getParameters().get('g-recaptcha-response');
        }
    }
    public String firstName{get;set;}
    public String lastName{get;set;}
    public String publicKey {get;set;}
    
    private String remoteHost{
        get {
            String ret = '127.0.0.1';
            //Also could use x-original-remote-host
            Map<String, String> hdrs = ApexPages.currentPage().getHeaders();
            if (hdrs.get('x-original-remote-addr')!= null)
                ret = hdrs.get('x-original-remote-addr');
            else if (hdrs.get('X-Salesforce-SIP')!= null)
                ret = hdrs.get('X-Salesforce-SIP');
            return ret;
        }
    }
    
    //Google Secret Key
    private static String secretKey = 'ADD YOUR GOOGLE RECAPTCHA SECRET KEY';
    private static String baseUrl = 'https://www.google.com/recaptcha/api/siteverify';
    
    //Constructor
    public GoogleReCAPTCHAController(){
        this.publicKey = 'ADD YOUR GOOGLE RECAPATCHA SITE KEY OR PUBLIC KEY';//Google Site Key or Public Key
        this.verified = false;
    }
    
    public PageReference submit(){
        if (response == null ){
            //Google recaptcha empty response
            return null;
        }
        HttpResponse res = getGoogleReCAPTCHAResponse(baseUrl,'secret=' + secretKey + '&remoteip=' + remoteHost + '&response=' + response);
        if (res != null ) {
            JSONParser parser = JSON.createParser(res.getBody());
            while (parser.nextToken() != null) {
                if ((parser.getCurrentToken() == JSONToken.FIELD_NAME) && (parser.getText() == 'success')) {
                    parser.nextToken();
                    this.verified = parser.getBooleanValue();
                    break;
                }
            }            
        }   
        if(this.verified){
            //Add your logic
            return null;
        }
        else{
            //Stay on page to re-try reCAPTCHA
            return null;
        }
    }
    
    //Get Google reCAPTCHA Service Response 
    private static HttpResponse getGoogleReCAPTCHAResponse(string requestURL, string body){
        HttpResponse response = null;
        HttpRequest req = new HttpRequest();
        req.setEndpoint(requestURL);
        req.setMethod('POST');
        req.setBody (body);
        try{
            Http http = new Http();
            response = http.send(req);
            System.debug('ReCAPTCHA Response-' + response);
            System.debug('ReCAPTCHA Body-' + response.getBody());
        }
        catch(System.Exception ex){
            System.debug('ERROR Message-' + ex.getMessage());
        }
        return response;
    }
}

Step 4: Create VF Page

<apex:page controller="GoogleReCAPTCHAController" sidebar="false" showHeader="false" cache="false" id="pg">
    <script type='text/javascript' src='https://www.google.com/recaptcha/api.js'/>
    <script type='text/javascript'>
        function recaptchaCallback() {
        var btnVerify = document.getElementById("pg:fm:pb:pbb:btnVerify");
        if (btnVerify.classList.contains("hideButton") ) {
            btnVerify.classList.remove("hideButton");
        }
    }
    </script> 
    
    <style type="text/CSS">
        .hideButton{
        display:none !important;
        }
    </style>
    <apex:form id="fm">
        <apex:pageBlock title="Google Captcha Example" id="pb">
            <apex:pageBlockSection columns="1" id="pbs"> 
                <apex:pageBlockSectionItem >
                    <apex:outputLabel for="fnField" value="First Name"/>
                    <apex:inputText value="{!firstName}" id="fnField"/>
                </apex:pageBlockSectionItem>
                <apex:pageBlockSectionItem >
                    <apex:outputLabel for="lnField" value="Last Name"/>
                    <apex:inputText value="{!lastName}" id="lnField"/>
                </apex:pageBlockSectionItem>
                <apex:pageBlockSectionItem >
                    <div data-type="image" class="g-recaptcha" data-sitekey="{!publicKey}" data-callback="recaptchaCallback"></div>
                </apex:pageBlockSectionItem>
                <apex:pageBlockSectionItem rendered="{!verified}">
                    <p>Google reCAPTCHA verified successfully.</p>
                </apex:pageBlockSectionItem>
            </apex:pageBlockSection>
            <apex:pageBlockButtons id="pbb" location="bottom">
                <apex:commandButton action="{!submit}" styleClass="hideButton" value="Verify" id="btnVerify"/>
            </apex:pageBlockButtons>
        </apex:pageBlock>
    </apex:form>
</apex:page>