How is a transmission line with a different characterisitc impedance shown on the smith chart?
A Smith chart natively represents normalized impedances/admittances relative to the characteristic impedance of the transmission line you're working on. In order to apply a mismatched quarter-wave section you'll need to convert impedances as you go down the line from load to source. 

For example, I have a source, 0.4λ 50-ohm segment, 0.25λ 100-ohm segment, and a 30+j40 ohm load. I'm interested in the impedance seen by the source as it drives this transmission line.

I take my 20-ohm load, plot it on a smith chart w.r.t. 100 ohm (i.e. at the point 0.3 + j0.5); I transform it through a quarter-wave to the point 1.2 - j1.6, which corresponds to the complex impedance 120 - j160 ohms. 

I'm now moving to the 50-ohm line, so I need to convert this to a normalized impedance w.r.t. 50 ohms: this is 2.4 - j3.2. By rotating through 0.4λ, I obtain the normalized impedance seen at the start of the 50-ohm segment: roughly 0.76 + j1.96. I finally convert this to ohms to obtain 38 + j98 ohms.