Were Airbus planes always fly-by-wire?
As Ron notes in the comments the A300 did not have it and neither did the A310 however the A300-600 and A310 did have electronically augmented controls according to the airbus site.  


  One of the A300-600 and A310’s notable innovations had been the
  introduction of electrical signalling on secondary flight controls,
  replacing the web of cables and pulleys tradionally used. Béteille
  wanted to take this evolution further with the next Airbus aircraft –
  to computer-driven digital “fly-by-wire”, in which the deflections of
  the flying control surfaces on the wing and tail are no longer driven
  directly by the pilots’ controls, but by a computer which calculates
  exactly which control surface deflections are needed to make the
  aircraft respond as the pilot wishes.


The A320 on had Fly-By-Wire

There is a nice little video on the same page that has some facts abut Fly-By-Wire tech over the years. A quick history note is applicable here: what we know of today as Airbus had heavy ties to the Concorde program which implemented the first commercially used analog fly by wire system and depending on how you look at it Airbus' first Fly-By-Wire system as well. These engineers who would later go on to work at what is now Airbus and drive the mentality into the designs which is why they have largely chosen it on all their airframes.