Is it possible to print a custom link to quick edit?

It depends. It's easy enough to trigger in-place editing of any given entity through code:

Drupal.edit.collections.entities.findWhere({entityID: 'node/1', entityInstanceID: '0'}).set('state', 'launching')

("in the collection of in-place editable entities, find the node entity with ID 1, the first instance of it on the page, and then launch in-place editing")

But then it's up to you to somehow determine which entity ID and which entity instance ID to use. Note that both of those values directly correspond to data- attributes in the DOM (data-edit-entity-id and data-edit-entity-instance-id, the latter of which is set through JS), so if there's always only one entity per page, you can easily have such a link! If there's multiple… then that's a UI problem for you to solve :)

What about "opening" state?
Can i set programmatically for quickedit "opening" state?
I try to do
Drupal.edit.collections.entities.findWhere({entityID: 'node/1', entityInstanceID: '0'}).set('state', 'launching').set('state', 'opening')
but there is error appeared.