Lethal damage and effects that say “destroy” won’t cause a creature with indestructible to be put into the graveyard. However, a creature with indestructible can be put into the graveyard for a number of reasons. The most likely reasons are if it’s sacrificed, if it’s legendary and another legendary creature with the same name is controlled by the same player, or if its toughness is 0 or less.

If you control any creatures as the triggered ability resolves, you must sacrifice one. You can’t choose to sacrifice Eldrazi Monument instead. You sacrifice the Monument only if you had no creatures to sacrifice.