We have a 2-year old heat pump and just today we notice the sound of gurgling water in our basement shower drain when the air conditioner runs. The sound is NOT heard in any of the other drains in the house.

What could be causing this sound?

The drain from the A/C unit is probably tapped into the shower drain pipe. No problem with that just make sure the A/C drain stay clear of all debris.

If you have a air vent tee installed before the P trap on the condensate drain it will make this noise or if a trap was not installed at all it will make this noise either or is wrong. The fan coil on a heat pump will try to pull air through the drain line when its running keeping it from draining. If a trap is not installed or installed wrong it can fill the drain pan and overflow with very little water going down the condensation drain.