After I take a shower, the hvac vent in the ceiling below the bathroom begins to leak. Does anyone have experience in fixing this problem?

The vent is leaking a fine stream (or constant drip). The restroom has not been used (consistently) in several years. And the HVAC unit has been idle for a little more than 30 days.

Is it possible that this will not continue after a few uses of the shower? Thanks for any commentation.

Chances are the shower is leaking and it just happened to find its way into the HVAC. Try several things to determine the actual leak site. I.E. pour a few buckets of water into the shower pan, if you get your leak it's the shower drain. Remove the shower head and fill several buckets out of the shower head, if you get a leak its likely the mixing valve or pipe leading to the shower head. If none of these produce a leak direct a shower stream onto the shower walls to check for bade chaulking. I am going to guess that your leak slowly comes to a stop about 5 minutes after the shower is over. Let us know what you find.
Bob

Bob's right... most likely nothing to do with the HVAC.

Just wanted to add...

You may also want to remove the shower handle and the trim plate on the shower and with a flashlight look inside the wall WHILE RUNNING THE SHOWER... see if can see any drips from above or at the valve itself.

Let us know what you discover... MARK