Did the author intend Farquhar's escape to seem believable?

Yes, the author certainly intended Peyton Farquhar's escape to be believable. That is the heart and soul of the story. The reader identifies with Farquhar and shares his thoughts and feelings in his imagination. The reader also has a "kinesthetic" relationship with the protagonist. He can feel the rope around his neck, feel himself falling, feel the shock of the icy-cold water, and all the other physical sensations involved in his escape from that terrible...

Yes, the author certainly intended Peyton Farquhar's escape to be believable. That is the heart and soul of the story. The reader identifies with Farquhar and shares his thoughts and feelings in his imagination. The reader also has a "kinesthetic" relationship with the protagonist. He can feel the rope around his neck, feel himself falling, feel the shock of the icy-cold water, and all the other physical sensations involved in his escape from that terrible situation. This is a marvelous conception and a marvelous piece of writing. The reader is closely united with Farquhar right up to the very moment when he reaches the safety of his plantation and is greeted with open arms by his loving wife.

Ah, how beautiful she is! He springs forwards with extended arms. As he is about to clasp her he feels a stunning blow upon the back of the neck; a blinding white light blazes all about him with a sound like the shock of a cannon--then all is darkness and silence!

What a rude awakening that is! Everything has been an illusion, and the whole fantasy of escaping and surviving has been imagined by Farquhar in the few seconds it has taken for him to fall from the bridge and for the six or eight feet of slack in the rope to play out. The strength of the illusion was the result of the strength of his desire to live, to be free, to get back to his home and to his loving wife.

In his cynical book The Devil's Dictionary, Ambrose Bierce defines "Hope" as:

Desire and expectation rolled into one.

It seems appropriate that an idealistic character like Peyton Farquhar should entertain such idealistic and unrealistic fantasies while falling through empty space during his last moment of life. He was easily taken in by the Federal scout's fabricated story because he wanted to believe it. The bridge seemed to offer the perfect opportunity for Farquhar to perform a dangerous and noble deed for the Southern cause. His dream of glory ended with the sudden shock of grim reality. The reader too is awakened simultaneously from the dream of home, peace, love, comfort, freedom, and all the good things of life.

Peyton Farquhar was dead; his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge.