Can I send my customer a price quote for a trip before they book, and turn it into a booking later if they say yes?

Yes, you can put a quote together before your customer commits, and turn it into a real booking later. You search as normal, pick the flights or the hotel you want to offer, add your service fee, and save it as a quote under that customer's name. It goes out to them as a neat proposal with your agency name and logo on it, so it looks like your own document rather than a copy of a search screen.

The one thing to be clear about is that a quote is not a held price. Airlines and hotel suppliers keep changing what they offer, so when your customer comes back a few days later and says yes, the system has to search again and the amount can come out different, or the room type you offered may have gone. Some fares can be held for a short period, but that depends on the airline or the supplier and on the plan you are on, so please do not promise your customer a locked price unless you have checked that it is genuinely on hold.

All your quotes sit in one list, so you can see which ones are still open, follow them up, and see which of your staff sent them. When the customer agrees you can carry the quote through to a booking without typing the passenger details again. Whether it goes through in one step depends on the same option still being available at that moment.

The best way to judge it is to build a quote for a trip you actually sell, send it to yourself first, and check that the layout, the wording and your fee all read the way you want before any customer sees one.