What is the relation between geodesic airframe and geodesic curves?
Before answering the questions, let's try to see how simple is the notion of a geodesic. In planar construction, we use latices to rigidify a structure, e.g. a truss:



The structure is made of straight segments that don't deform when axial forces are applied. If we switch from the plane to an arbitrary curved surface, something similar can be done, but geodesics take the place of the straight lines (actually the line is the geodesic for the plane).

The idea behind a geodesic construction is that its rigidity is obtained from segments on the surface of the truss, and the inner volume is left empty and available for other uses.



What is the relation between geodesic airframe and geodesic curves?

Are geodesic frames related to geodesic curves? If yes, in what way?



Geodesic airframes are indeed constructions made of geodesics, a technique first applied to aviation by Barnes Wallis. From the Barnes Wallis Foundation, and article Geodetic Aircraft Design:


  A geodetic (or geodesic) airframe makes use of a space frame formed
  from a spirally crossing basket-weave of load-bearing members. By
  having the geodesic curves form two helices at right angles to one
  another, the members became mutually supporting in a manner that the
  torsional load on each cancels out that on the other. In addition to
  being comparatively light and strong, the fact that the geodetic
  structure was all in the outer part of the airframe meant that the
  centre was a large empty space, ready to take payload or fuel.



A close-up view of the geodetic structure designed by Barnes Wallis (source)


Vickers Wellington's under construction (source)



[The wikipedia article] does not seem to deal with  geodesic curves. It is mostly about about a structure made by triangles or other polygons



Geodesics are curved lines. Using 2 or 3 geodesics on a closed surface creates repetitive and symmetrical curved polygons, e.g. curved triangles and curved lozenges. Providing the supporting geodesics are angled so that "the torsional load on each cancels out that on the other", the resulting geometry belongs to the family of geodesic modeling.

Whether the actual construction is an approximation of the perfect model is a different engineering/costing problem. This is related to the actual cancellation of the torsional loads after approximation and the effect of the residual loads on the structure. This residual load could potentially be absorbed by additional engineering device if appropriate. This dome is made of straight segments only, the triangles are planar.


The geodesic dome at Amundsen Scott South Pole Station by Ernie Mastroianni (source)