No.
You can just count how many times the user has clicked your print button. If this is enough: the same way you increment the modal open counter, you can increment the click counter of the print button. You can do something like this:
HTML:
<button id="botaoPraImprimir" type="button">Imprimir</button>
<input type="hidden" id="contadorDeImpressoes" value="0">
Javascript, with jQuery:
var $contadorImpressoes = $("#contadorImpressoes");
$("#botaoPraImprimir").on("click", function () {
let valorAtual = parseInt($contadorImpressoes).val();
$contadorImpressoes.val(valorAtual + 1);
});
Note that this will count how many times the print button of your modal has been printed. So you know how many times the user had the intention to print .
The print button in the browser's print dialog is not accessible via code - and there is no browser print event that you can capture or intercept. You can not count how many times a document was actually printed , or how many copies were made .