Can I refund a flight ticket for my customer through the same system I booked it in?

Yes, in most cases you can start the refund from the same booking screen you used to make the sale, although how much of it happens automatically depends on the supplier sitting behind that fare and on the ticket rules.

Where the flight booking API allows it, your staff open the booking, cancel the ticket and raise the refund request in one place. The system passes the cancellation to the airline or the consolidator, and the amount that comes back is worked out from the fare rules attached to that ticket, after any airline penalty and after whatever service fee your agency chooses to keep. The booking record then shows as cancelled or refunded, so your sales reports and your accounts stay in step with what actually happened.

Be aware of the limits. Some fares are simply not refundable, and on those the most you can recover is usually the taxes, if the airline permits even that. Many tickets can be voided free of charge only within a short window on the day of issue, and after that they move to the slower refund route. A number of suppliers do not offer a fully automatic refund at all, so the request goes to a support desk and is settled offline. The time it takes for the money to reach you, and for you to pass it to your customer, is set by the airline and by the payment route, not by the software.

It is worth asking each of your suppliers which of their fares can be cancelled online, then testing one low value booking end to end so your team knows exactly what to expect before a real customer is waiting.