Some people use serger thread to quilt. However, most quilters don't want to use it.
If you look at it closely you will see that it is quite fuzzy compared to regular sewing thread. This fuzz creates a lot of lint. People who use serger thread to quilt typically clean the bobbin area at every bobbin change to keep the machine clean. Lint in a machine is bad news in terms of long-term wear, servicing requirements, etc.
Another reason to be wary of serger thread is because it is a weak thread that breaks more easily than regular sewing thread. This would be a problem especially if your quilting lines are far apart, which puts more stress on the thread during normal wear and tear.
Finally, if the small quilting machine you are talking about is high speed (as most frame quilting machines are), you are likely to experience a lot of thread breakage while quilting with serger thread. High speed machines require thread with higher tensile strength than serger thread.
So, yes, it's possible to use serger thread to quilt with (as long as it doesn't keep breaking on you), but there are reasons to choose better quality thread for quilting.