The Greek text is:
μέγαν ὄκνον ἔχω καὶ πεφόβημαι
πτηνῆς ὡς ὄμμα πελείας.
Sophocles. Aias, lines 139–140. Perseus Digital Library.
ἔχω = I bear/carry
μέγαν ὄκνον = great fear
καὶ = and
πεφόβημαι = I have been alarmed/terrified
ὡς = as/like
ὄμμα = eye
πτηνῆς πελείας = of winged dove
So, literally, “I bear great fear and have been alarmed like the eye of a winged dove.” The rhetorical figure is synecdoche (part standing for whole): it is the dove that has been alarmed, not just its eye. There is a tradition of translating this figure into English by describing the motion of the dove's eye: Jebb has “troubled eye”; Storr has “quivering eye”, and Scully, quoted in the question, has “quick, scared, look”.
The reason for using a more specific simile (“doves” rather than “birds”) is to create a more specific image in the imagination of the audience: not all birds behave in the same way. Doves, in particular, are too large to hide in foliage or undergrowth, and do not have the weapons to fight back, so their only resources against predators like hawks and foxes are their quick reactions and swift flight.
Lima (1993) … classified patterns of escape as follows: into woody vegetation (mostly passerines), into herbaceous vegetation (marsh and grassland birds), around the back of tree trunks (woodpeckers, creepers, and nuthatches), or into the air, here distinguishing between speed-based tactics (doves and Galliformes [fowls]), aerial dodging (larger gulls, owls and corvids), and socially coordinated escape (Charadriiformes [shorebirds] and Passeriformes [songbirds]).
Tim Caro (2005). Antipredator Defenses in Birds and Mammals, p. 419. University of Chicago Press.
If you spend any time watching birds then you’ll become familiar with these behaviours, as Sophocles and his audience surely were.