Interesting button. The first thing to say is that the “Ld” at the end of the Firmin name is for “Limited” and Firmin didn’t become a limited company until 1875, so it’s after that date. Note that it’s “Firmin & Sons” [plural], not “Firmin & Son” [singular], which is the usual format for this period.
I would assume it to be a livery button. The crests appear to be a griffin passant left, an indeterminate animal head erased, and an arm erect holding a dagger. I can’t be sure about the animal because it’s somewhat corroded. I thought it might be a hind (female deer), but the snout is a bit short and there’s some patterning on the lower neck suggestive of something shaggy/furry… so I wouldn’t rule out wolf, despite the rather long neck.
The surround is, as you say, for the Order of the Garter with its motto: “Honi soit qui mal y pense” (from Middle French, meaning “Shame on him who thinks evil of it”). It’s the oldest and most senior Order of Chivalry in Britain but I wouldn’t read too much into it, except that it means the button likely relates to nobility with a family head entitled as “Sir…” There’s no Royal crown above, so no direct connection to Royalty. It’s difficult to make out exactly what that is at the top of the garter but it looks be a demi-eagle with wings spread, issuant from a marquis’ (?) coronet. Perhaps something emblematic either side of the eagle too.
It's not unusual for livery buttons to have two crests, arising from intermarriage between families, but three is somewhat unusual (four, if you count the crest at the top of the garter). The trouble is that they’re relatively common heraldic emblems with two of them uncertain and I think it would take some considerable searching in Fairburn to track down who the button relates to.
