I'm reading it as they used the IBM AT as a serial terminal, running QMODEM as the terminal emulator, for a Linux host that was connected to the Internet via modem.
BUT, it was definitely possible to do what you're describing with some combination of a dialup shell account, a terminal program like qmodem and something like https://en.wikipedia.org/wiki/Slirp
I agree it's highly unlikely that the AT was running slirp. Wikipedia says an AT was a 286, so it wouldn't have been linux. Not even sure what the options would have been. Minix? Xenix?
I can confirm that they did run Minix OK, although I remember the network support was iffy at best. We never got it to work at any rate. XENIX would have been hard to get your hands on. I think QNX would run on an AT as well, although my memory might be playing tricks on me there.
Having the IBM AT a a serial terminal would let somebody run CLI-based software on the Linux box (like Lynx, an IRC client, FTP, etc). You'd just be using a shell account on the Linux box.
I did stuff like this in the early 90s at home and later at a company I worked at (sharing a single dial-up connection over 10Base-2 with 5-ish Windows 95 PCs).