You can set a field in your document to a server timestamp which tracks when the server receives the update.
import { updateDoc, serverTimestamp } from "firebase/firestore";
const docRef = doc(db, 'objects', 'some-id');
// Update the timestamp field with the value from the server
const updateTimestamp = await updateDoc(docRef, {
timestamp: serverTimestamp()
});
// Create a document reference
const docRef = db.collection('objects').doc('some-id');
// Update the timestamp field with the value from the server
const res = await docRef.update({
timestamp: FieldValue.serverTimestamp()
});