Once you are in screen space coordinates it should be pretty straightforward to check if either at least on of the quad's points lie within the screen area, or if any of the four screen points lie within the quad (this can be done with at max 12 dot products in 2D). Last one is needed if only part of one edge of the quad is onscreen. That should be it.

You can do the same thing in world space, using distance to plane (points inside view frustrum) and line-triangle intersection tests (frustrum edges against quad).