Can my travel agency software show prices in my own currency when the supplier quotes in a different one?

Yes, travel agency software can normally show prices in the currency you and your customers work in, even when the supplier quotes in another one.

The way it usually works is that the supplier returns the fare or the room rate in its own currency, and your system converts it before the price is displayed. Some setups use a rate that the software refreshes on its own, and others let you set the rate yourself so you stay in control of what your customers see. You can normally also add a small buffer on top of the converted amount, so that a shift in the rate between the search and the payment does not eat into what you earn.

There are limits worth knowing. The currency your customer is charged in is not always the currency the supplier bills you in, so your settlement can still happen in the original currency, and your bank may apply its own rate and charges on top. Some suppliers also insist that the booking is paid in their currency, which means the converted amount your customer sees is only a guide until the payment actually goes through. How many currencies you can offer, and whether conversion is available at all, depends on the supplier, the API and the plan you are on.

It is worth deciding early which currency you want as your base one, then checking with your payment gateway that it accepts every currency you plan to display, because a converted price is only useful if the money can be collected the same way.