Forked fingerings are usually when fingers 1 and 3 are used together on either right or left hand so the sounding note issues from the tonehole left open in between them - only on some altissimo notes are true forked fingerings used. Forked fingerings are mostly found on earlier instruments with only a few keys and simple system instruments such as recorders, Baroque flutes and oboes, Classical era clarinets, fifes and they're also still employed on simple and modern German and Oehler system clarinets, German bassoons and modern oboes - Boehm system flutes and clarinets have pretty much eradicated forked fingerings and as saxes are largely based on the Boehm principle, they too have pretty much eradicated forked fingerings.
F# on saxes when played with the cross key isn't (by definition) really a forked note, even though it's played with RH 1 and 3 as the F# issues from the tonehole at the back (above the thumbhook) and there aren't any toneholes closed below it (the E and C# toneholes are too far away to count), whereas F# with the normal fingering (xxx|oxo) is in essence a forked note as there's a tonehole closed below the one the F# issues from. Likewise with C when played with LH2 only (oxo|ooo). Saxes fitted with the forked Eb mechanism do use the forked fingering (xxx|xox), but again the Eb produced this way isn't by definition a true forked note as it's issuing from a tonehole near the usual Eb tonehole with another tonehole open below it and not issuing from a tonehole that's left open between RH fingers 1 and 3.
Another meaning of forked keys refers to the connection or linkage between two parts of a mechanism, such as the side Bb and C keys which are linked by forks at the ends of the key arms to the projecting pins on their respective pad cups (eg. Yamaha saxes up to and including the 82Z, Selmer SBA, SA80II and SIII and Yanagisawa saxes among many others). The front F touchpiece on Yamaha 62 sopranos has a fork linkage in it to connect it to the rocker that lifts the high F key.