"Ação de despejo" refers specifically to an action seeking to force someone to leave the property. "Propor uma ação de despejo" can be translated as "to file for an eviction", "to file/commence an action for eviction" or "to file/commence an eviction lawsuit".
There are multiple definitions - click on the translation that matches your context.
the act of forcing someone to leave the place they are living in, especially using the law, usually because they have not paid their rent; the act of forcing someone to leave somewhere
Hemsworth Town Council has told tenants to remove cockerels from the site or face eviction.