Depending on how traditional the band is, it may be a little difficult to employ quartal harmony. Since quartal harmony is definitively different than triadic harmony, you're essentially try to use something that doesn't "belong". Using quartal harmonies on top of triads can cause some major clashes and/or result in the harmony being changed. For instance, quartal harmony in place of a major triad will potentially cause a clash with the major third of the initial harmony, so if someone else is still playing that major chord and/or the third appears in the melody, you will either end up creating a minor second or b9 between the third and the fourth, which is usually more dissonance than traditional folk calls for, or if the fourth appears lower than the third, it could change the chord and/or its function.
Using quartal harmony in place of triadic harmony was used by Miles Davis on "Kind of Blue". Part of the reason he was able to do this so effectively is because he was composing in Jazz, where all of the chords allow for extensions, unlike folk that typically uses triads and 7 chords. Davis stacked chord tones quartally but was still using triad based harmony and you will actually find that there is a major third on top of those chord shapes, so it's not strictly quartal. It does still supply the feeling of quartal harmonies though, so I've never really heard anyone argue it.
So trying to use quartal chords in place of triads could be possible but it's going to actually change the harmony instead of being another way to utilize the triads. You can't really use quartal voicings for triads because the only fourth within them is the fifth up to the root. If you want to try to do something like that, you have to include additional notes, like extensions and alterations, or you would need to rewrite the harmony entirely, taking care to avoid the dissonances that can come from the minor 2nd/b9 interval that I referred to earlier.
Todd mentioned in a comment that there are some chord types that are kind of quartal in nature, such as the sus4 chord, but they do function differently and are an extension of the triadic system. You could end up using this as a means to convey the chord types to those that don't understand the quartal harmony concept too, as lots of musicians are aware of sus chords.
It's hard to say what would be the best way to approach this with the given chord progressions you've presented due to the intrinsic differences between triadic and quartal harmonies. You could try using extensions and see if it works out with the below suggestions.
For a major chord, you could try your quartal voicing starting with 3 as the lowest note and stack 4ths: 3, 6/13, 2/9, 5, 1 (for a C major chord: E, A, D, G, C)
For a minor chord, you could try starting at the fifth: 5, 1, 4/11, 7, 3 (for an A minor chord: E, A, D, G, C)
For a dominant chord, which will have a different sound than most due to its tritone, you could start on 7: 7, 3, 6/13, 2/9, 5, 1 (for a G7: F, B, E, A, D, G)
As you can see, these would all have to include additional notes to be able to accommodate 4ths the whole way up. This can add some "richness" to your harmony but it's not something you really see in traditional folk. But you could come up with some great sounds and try to bring some progressiveness to your traditional group if there's room for you to make such a suggestion.