<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>Title</title>
<script src="https://code.jquery.com/jquery-3.0.0.min.js"
integrity="sha256-JmvOoLtYsmqlsWxa7mDSLMwa6dZ9rrIdtrrVYRnDRH0=" crossorigin="anonymous"></script>
<script src="//www.google.com/recaptcha/api.js"></script>
</head>
<body>
<div class="recaptcha"></div>
<button class="button">Кнопка</button>
<script>
i = 0;
$('.button').click(function () {
if (i % 2) {
$('.recaptcha')[0].innerHTML = '<div class="g-recaptcha" data-sitekey="6LcXdm8bAAAAAIfagC-DXbNuWUiEstmlfG04eskc"></div>'
} else {
$('.recaptcha')[0].innerHTML = ''
}
i++
//console.log(i)
})
</script>
</body>
</html>