That's something different. You may be thinking of Cepheid Variable Stars, which as it turns out have a closely linked period of brightness variation to absolute brightness. I don't think there's many who question the perodicity/brightness connection of Cepheid variable stars.
The article is referring to detecting the Doppler shift of spectra in the light from an object in space, and using that to infer if the object is moving away or towards the observer. It's a theory that is easily observable in many areas of nature. There are two leaps of faith coming into play here. First is when the observer assumes that because an object appears to be moving away from the observer (based on its spectra), that it in FACT moving away from the observer, and second is assuming that because an object is moving away from you now, that it has always been moving away from you. Really cool stuff! As for me, I've never been completely convinced that a high red/blue shift is absolute proof of a high relative velocity, but it appears to have fit well with observations up to this point. This'll be interesting to follow up on.
Yes, but the relationship between redshift and distance is what's at issue here. We know that certain galaxies are redshifted, but how do we know that they are far away? We know it by looking at the Cepheid variables. At some distance, you lose the ability to distinguish individual Cepheids, but fortunately there's enough overlap between the measurably redshifted galaxies and the distinguishable-Cepheid galaxies to get a measure of the Hubble constant. Once you have that, you can plug in redshift and crank out distances--but not, it seems, for quasars!