As all pervasive, stationary fields, do the QFT fields (various Quark, electron, EM, etc.) constitute the modern equivalent of the traditional ether concept?
Ether is a somewhat vague term that is used in different ways. The point of the ether originally proposed to explain the propagation of light is that it constitutes an absolute rest frame. So for example you would know if you were moving relative to the ether because you would measure different velocities of light in and opposite to your direction of motion. If this is what you mean by an ether then quantum fields do not constitute an ether since they are Lorentz covariant.
If by ether you just mean something that is present everywhere then things get more complicated. For a start the objects physicists mean by quantum fields are not physical objects. A quantum field is an operator field i.e. it is a field whose value at all points in spacetime is a function. So the quantum fields are purely mathematical objects not physical ones. You could argue that the quantum field must be a description of some physical object, and that this physical object would be the candidate for an ether, but the physical interpretation of quantum fields is currently mysterious.