Can I be sure the fare I see in the search results will still be the same when my customer goes to pay?

Usually yes, but not always. The fares you see in a search are the ones the airline was offering at the moment that search ran, and an airline can move a price or sell the last seats at that price while your customer is still deciding. Most bookings are quick enough that nothing changes, so in day to day work the fare you quote is the fare your customer pays.

Before the booking is confirmed, a flight booking API checks the fare again with the airline. That check tells you whether the fare is still live, and if it has moved it gives you the new one before any money is taken. This is where a change shows up, not after the ticket is issued, so your customer is never charged something they did not agree to.

A price is more likely to move when the route is busy, when only a few seats are left at the cheaper fare, when the fare is part of a short sale, or when a lot of time passes between the search and the payment. Taxes and surcharges can also be worked out again at that final check. How often this happens depends on the airline and on the supplier behind the fare, so nobody can honestly promise it will never happen to you.

Keep the gap between searching and paying as short as you can, and always show your customer the confirmed fare from that final check before you take their money.