If one of the players successfully performs a "Hunter's Quarry" action (or maneuver) on one target. Does this mean that other players benefit from this action too? Which means that they upgrade their attack against this target once?

Here is a corresponding part of the rule text: "[...] If the check succeeds, upgrade the ability of all attacks made against the target by one until the end of the character's next turn."

Yes.

First because it says all attacks, not all attacks by the character, and also because with the Hunter's Quarry action, upgrading an attack next turn would be a pretty small benefit from losing an action.

Plus otherwise it would compare very badly with True Aim, which upgrades an attack, in addition to normal benefit, from the aim manouver for a character, and has multiple ranks!

What can you do (as a GM) with all the success, advantages, disadvantages, and other results form the dice result of the hunters quarry action?

Maybe have the first attack ignore a defense point or add one for themselves, gain an advantage/threat to their next check, have the target/attacker lose cover.