Yes, in many cases you can change the date on a ticket through the API, but it depends on the airline and on the fare your customer bought. The API lets you send a change request against an existing booking, and it comes back telling you whether a change is allowed, what the new options are, and what the change will cost. Cheaper fares are often sold with no changes permitted, and for those the only route is to cancel, if the fare even allows that, and book again.
Where a change is allowed, the usual flow is to ask the API what the fare rules say, then request the new date, then confirm it. Your customer normally pays the difference in fare plus whatever the airline charges for making the change. The API shows you both parts before you commit, so you can tell your customer the full amount and get their agreement before anything is confirmed.
There are limits worth knowing. Some low cost airlines and some suppliers do not support changes through the API at all, even when the same airline allows a change on its own website, and in those cases your team has to go to the airline or the supplier directly. Changes very close to departure, or after check in has opened, are often refused. A change of passenger name is a separate matter again, and most airlines do not permit it on a ticket that has already been issued.
Before you promise a date change to a customer, pull up the fare rules for that booking through the API and read what they say about changes, then quote the amount the API gives you rather than an amount you expect it to be.