Will an FPGA get damaged if contention occurs on a bidirectional pin on it?
Just like you were told. If one device tries to pull high whilst the other tries to pull low the current will flow at will up to a level determined by the silicon resistance of the two output structures on the devices. Some devices may have current limit designed into them but even then the current is very likely to be way higher than the specified safe limit for the parts. 

High current flow like this can quickly build up heat to a level that destroys the IC chip in the area around the pin connection. In some cases the failure could destroy the whole chip. In others there may be parts of the IC that remain functional with loss of some pins functionality.

If you are trouble shooting a board that was just built and has a contention issue due to a wrong connection or solder short it is often possible to fix the bad connection and have the board function normally. However it is always possible that the affected devices will have been stressed some and they could fail at some later time. Metal migration on chip could happen to create a short or stress could have damaged an oxide so that it fails when there is a transient at power on or power off time.