I think you are confusing "quarter-wave matching" and "source termination" (or "series termination", if you prefer).
First, a terminology issue: It may just be me, but when you refer to R of a transmission line, I think of R from an RLGC model of the line. From context, I can tell you are referring to the real part of Zo. But unless you are dealing with non-TEM modes in cutoff, Zo is mostly real so most people just use Zo not Rxxx. At least this is my experience, others may disagree.
Getting back to your original question, there are many different types of matching. For RF, there is also single and double stub, and for PCB's there is parallel or load terminations. In source termination, the source is matched to the transmission line, i.e., output resistance + R = Zo. Waves are reflected at the load, travel back to source, and are absorbed in the source resistor(s). In an RF application we don't want energy absorbed by the source, we want it absorbed by the load. A quarter-wave transformer maximizes the power absorbed by the load at a given frequency.