Can I hold a hotel room for my customer for a few hours before they pay for it?

Not usually. Most hotel rates that come through a booking API are either confirmed straight away or not at all, so the room is only yours once the booking is actually committed. Some suppliers do allow a hold, sometimes called a provisional booking, where the room is blocked for a set period and you confirm it later, but that is the exception rather than the normal way hotel rates work, and it will only be offered on the rates where the supplier permits it.

What you can always do is save the search result and send your customer a quote. A saved price is not a held room though. Both availability and price can move between the moment you searched and the moment you book, so if the customer comes back later in the day that exact rate may be gone or may cost more. The system rechecks the rate before it confirms the booking, so you will not end up selling a room that no longer exists, but you may have to go back to the customer with a new price.

Many agencies get around this by looking for a rate that can be cancelled free of charge up to a certain date. You book it properly, so the room is genuinely held in the guest name, and if the customer changes their mind you cancel inside the free window. That gives you a real hold rather than a promise, although you do have to pay or use your credit with the supplier at the time of booking.

The simplest next step is to ask which of the suppliers connected to your account support provisional bookings, and in the meantime get into the habit of checking whether a free cancellation rate covers what you need.