Reach for Dulux Highland Mist. It lands at ΔE 1.1 from Farrow & Ball Pigeon, which is well inside the "very close" threshold (under 2.5) and nudging towards imperceptible. With an LRV of 35.3 it sits in the same mid-tone bracket as Pigeon, so you'll get that same soft, muted grey-green that makes Pigeon such a quiet, easy colour to live with. Side by side under decent light you'd struggle to call which is which.
The other contender Dulux throws up is Flint Arrow (LRV 29.3), but at ΔE 4 it's a noticeably different beast — darker and a touch more sombre. It's a perfectly nice colour, just not a Pigeon match. If anyone tells you Flint Arrow is "basically Pigeon", they've not held the two up together. Highland Mist is the one.
Now, the honest bit. Pigeon's charm is partly down to how Farrow & Ball builds depth into the colour — that complexity in the pigment is what gives it the chameleon quality where it reads green in one light and grey in another. A Dulux match nails the hue and the depth admirably, but the very best way to keep that F&B character is to spec it in Dulux Heritage, which has a richer, flatter finish that flatters muted mid-greens far better than standard trade matt.
So: if budget's the driver, Highland Mist in a quality Dulux finish will do you proud. If you're matching to existing Pigeon woodwork or furniture, take a brushout — never trust a screen or a fan deck swatch for a colour this subtle. Paint an A4 patch, live with it across a full day, and check it against the real Pigeon before you commit a whole room.