In Elder Sign, does the "Bind Monster" card avoid an Ancient One effect?
No, it does not include the Elder Sign that must be lost per Yig's abiliy

The rules are not very clear on this matter:


  Defeating Monsters with Items and Spells - Some items or spells instruct the player to defeat a monster. When using such an item or casting such a spell, the player chooses any monster in play and flips its marker facedown, ignoring all of its tasks. If the monster marker is in a partial monster task, the uncovered task requirements must still be satisfied in order to resolve the adventure. After either resolving or failing the adventure, the player collects the defeated monster marker at the end of his Resolution phase. (page 17)


Tasks are ignored (including costs), which might lead you to believe that Yig's Elder Sign loss wouldn't occur, but the FAQ says:


  What do items mean when they say “defeat” a monster? - When a player uses an item/spell that “defeats” a monster, that player chooses a monster in play (not necessarily at the same adventure his investigator is at) and turns that monster facedown. None of the monster’s costs must be paid (such as advancing the clock or losing sanity/ stamina) and all of that monster’s tasks are considered to be completed. The monster is collected as a trophy when the player’s turn ends (and not before), whether the investigator succeeds at the adventure or not. Note that if a monster is only partially covering a task, any uncovered parts of the task are still uncompleted for purposes of succeeding at the adventure. (page 5)


The operative words there are "considered to be completed." Yig would trigger one time for each task on a Cultist monster defeated by Bind Monster, causing the investigators to lose one (or more) Elder Signs.


  Each time an investigator completes the task on a Cultist monster marker [...] the investigators lose 1 Elder Sign, if they have any."