No — a message you send does not trigger an instant alert on the other side, and read receipts do not work today.
While a conversation is open on screen, incoming messages appear on their own; you never have to reload the page. If the other side has the screen closed, nothing reaches them by email or phone: they see your message the next time they open the demand and go to Messages.
The check mark next to a message you sent stays on Sent for good. Delivered and Read are never produced, so the app cannot tell you whether the other side has read you. For the same reason the unread counter in the conversation list always shows zero, there is no online/offline signal for the other party, and there is no "typing…" indicator.
If something is time-critical, do not rely on chat alone — also put the detail into the demand or the offer itself, which is where the other side looks while working on the quote.
Also asked as: Does my message reach the other side instantly? · Are there read receipts (blue ticks)? · Do you send a notification when a message arrives? · Can I see whether the other party is online?