Can I take the airport train straight from Hongdae to Incheon Airport, or do I need to change trains?

Yes — you can ride straight there with no transfer. Hongik University Station (Hongdae) is a stop on the AREX All-Stop Train, so you can board right there and stay on until Incheon International Airport.

Fare: ₩4,500 (as of today)
To Incheon Airport Terminal 1 Station: about 54 minutes after boarding
To Incheon Airport Terminal 2 Station: about 59 minutes after boarding

Check which terminal your airline uses before you set off. Terminal 2 is the last stop, so if you're flying out of Terminal 1, get off one stop earlier.

Tip: Hongdae is served by the All-Stop line only. The AREX Express Train runs non-stop from Seoul Station and doesn't stop at Hongdae, so just take the regular All-Stop train.