Brief response: No
The browser alert is a simple DOM alert, as described in the W3C "show an alert to the user and wait for it to close"; as a rule: there is no specification, although browsers can even let you customize how they do with scrool bar: link but this would be out of specification
How to customize (Simple alternative)
You need to make your own alert and imitate browser behavior by javascript.
example:
JavaScript:
<script>
function customAlert(message) {
var div = document.createElement("div");
div.classList.add("custom-alert");
var close = document.createElement("a");
close.textContent = "[x]";
close.classList.add("close"); // para o seu css
close.addEventListener("click", () => {
document.body.removeChild(div);
}, true);
div.appendChild(close);
div.appendChild(document.createTextNode(message));
document.body.appendChild(div);
}
customAlert("ola");
</script>
CSS
.custom-alert {
/* o css do seu alerta */
}
W3C dom alert: link