Yes, TG will be fined for this. Penalties in Taiwan can be quite severe, including the possibility of having landing rights revoked for several days.
I am surprised that they care about the passport validity though. I've never seen most requirements enforced (return tickets and passport validity are ones I've rarely heard of being enforced). I had a family member work as a CSR at UA. Once they had someone checking in for a flight to AMS but their passport didn't have the 6 months on it. They wound up calling people in AMS and were told the validity didn't matter. And I've heard of people getting deported from Brazil. Although in some places they fine you too. Lots of stories from UA about people getting deported (and other stuff people do to avoid getting deported).