Of course. Create a script content specific to your site in the extension and add an invisible marker in the DOM, for example:
$('body').append('<div style="display: none;" class="extension_enabled" />');
Set a short timeout on the page to check this after the document has fully loaded, for example:
$(function() {
setTimeout(function() {
if ($('.extension_enabled').length > 0) {
alert('Installed!');
} else {
alert('Not installed.');
}
}, 500);
});
NOTE: The code is in jQuery format for simplicity. Of course you can do this with raw javascript.