How to send email using javascript on Salesforce Visualforce Page?

Yes you can do with Salesforce AJAX Toolkit. You can write something like this in your JavaScript. You can update the SendMail function with all the required parameters and call from the onclick action of the button.

<apex:page>
    <script type=”text/javascript”>
        var __sfdcSessionId = ‘{!GETSESSIONID()}’;
    </script>
    <script src=”../../soap/ajax/34.0/connection.js” type=”text/javascript”></script>
    <script type=”text/javascript”>
        function SendMail() {
            try {
                sforce.connection.sessionId = “{!$Api.Session_ID}”;
                var message = new sforce.SingleEmailMessage();
                message.replyTo = “xxxx@gmail.com”;
                message.targetObjectId = “xxxxxxxxxxxxxxxxx”;
                message.templateId = “xxxxxxxxxxxxxxxxx”;
                message.saveAsActivity = false;
                message.plainTextBody = “this test went through ajax”;
                var result = sforce.connection.sendEmail([message]);
                if(result[0].success == ‘true’) {
                    alert(“Mail sent successfully”);
                } else {
                    alert(“I had a booboo.”);
                }
            }catch(e) {
                alert(“Error Occurred: nn” + (e.message||e));
            }
        }
    </script>
</apex:page>