It could connect you to a machine that had Internet access. Some ISPs offered that as a service (you'd get some kind of BBS-like interface or - if you were lucky - a UNIX shell), but that's not the same thing.
QMODEM was essentially just a terminal emulator that used a serial port and understood how to control a modem.
After all this discussion here my conclusion is that "connecting to the internet" is an ambiguous term.
It can mean "have IP connectivity, i.e., IP packets routed to and from the internet" in which case the described PC was not "connected to the internet with QMODEM". It didn't do anything IP.
It can mean "have a terminal that can interact with information from/to the internet" in which case the PC was indeed "connected to the internet with QMODEM".
To me, the second meaning is quite the stretch, but apparently to others it's fine.
My problem with your definition is that it doesn't take into account the reality of connectivity at the time (at least in my experience of the early 90s) - not a whole lot of machines had IP stacks that were connected via ethernet/isdn/t1/etc and online all the time. Certainly you'd have to be pretty special to actually own one or have one at home. Connecting over some kind of tty or dialup was extremely common.
So using your definition, a sizeable percentage (possibly even a majority?) of people who were online and doing things on the internet during the QModem era were doing it through computers that were not "connected to the internet". Which seems obviously silly.
> that's not the same thing.
I think your definition of "connect to the internet" make sense today, but would be ridiculously narrow when applied to the QModem era given the computing landscape at the time. Where do you draw the line? Using a tty style terminal connected via serial to a unix box connected via ethernet? How about SLIP/PPP?
I guess my problem with your definition is that you end up saying that a very large percentage of people who were online at the time were using the internet through computers that were not "connected to the internet".
Until the mid-90s the internet was predominantly text anyway, so it's not like you were missing out on a whole lot if you were "only" using a terminal.
However, the user running QModem was on the Internet.
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).
The two Linux box were one with the modem and the other via Nat (Ethernet but with old coaxial cables). The AT was just a terminal.