Green's the easiest colour to live with because it's drawn from nature, so almost anything that exists alongside it outdoors works on your walls too. The trick is deciding whether you want a calm, tonal scheme or a bit of contrast.
For a soft, restful room, partner green with warm off-whites and creams. Farrow & Ball Au Lait (LRV 80) is a lovely milky cream that takes the edge off a deeper green without going stark, and Paper III from Paint & Paper Library (LRV 75.3) does a similar job — both are bright enough to lift ceilings, trim and woodwork while keeping the whole thing gentle. This is the safe, never-fails route, especially with mid-to-deep greens like Card Room Green or Calke Green.
For warmth and depth, reach for browns and russets. Mylands Cigar BH.20 (LRV 11.8) is a properly rich tobacco brown — fantastic on a chimney breast, a piece of furniture or as a grounding accent against green. Green and brown is the woodland pairing; it always reads natural and grown-up.
For a bit of life, green's true complement on the colour wheel is red-pink, which is why a dusty pink or fuchsia accent sings against it. Dulux Fuchsia Falls 2 (LRV 29.8) is a confident pink that brings energy — use it on a single piece, soft furnishings or as a feature rather than a whole wall, unless you're going all-in.
The "but what about" question: what trim colour? With green walls, go off-white rather than brilliant white — pure white looks clinical next to natural greens. Pointing or Slipper Satin keep things soft.
Practical advice: pick your green first, then test your partners on a big sample card against it, in the actual room light. Green shifts a lot between north and south light, and a pairing that sings at noon can fall flat by lamplight.