Does anyone know any pieces of roughly grade 4 standard for a string quartet? Thank you in advance

Might be a little hard, but what about the "Entrance of the Queen of Sheba". I'm only about grade 3/4 and can manage 2nd violin fine in it.

Try the book 'Easy String Quartets' Book pub by Faber.
Also Charles Hambourg String Quartet No 2 in F (pub Lengnick/William Elkin)
Locke Suite No 3 in C (Peters)
Johann Fischer Tafelmusik Suite (Richter)

This may sound difficult, but have a go at Pachelbel's canon - except you would probably want to play it extremely slowly. Most of it isn't actually that hard, and i know someone that could play it at approximately grade 3/4 standard.

Probably a bit hard to perform at grade 4 level but good to have a go at for fun.

Haydn String Quartets Op 1 No3 , Op 9 No 6 are a couple of the easier ones I think

Mozart K80, K156(there's a recording of a bit of this played by forum members on the forum recordings website I seem to remember) K157, K168, K169

try going on http://www.mutopiaproject.org

Ragtime, Serenade and Rhumba (or is it Rumba? I forget) by Anita Hewitt-Jones is one of my all time favourite intermediate string quartet pieces. I should think it'd be OK for G4ish - been a long time since I played it. Attractive tunes, accessible, fun.

I don't think Pachelbel's Canon is grade 4 for the cellist! Chaos, ask your cello teacher - she might have some ideas.