Use the search query string parameter to perform a case-insensitive search within all designated text fields of a model (added in AWX 1.4):
http://<server name>/api/v2/model_verbose_name?search=findme
Search across related fields (added in AWX 1.4 / automation controller 3.1):
http://<server name>/api/v2/model_verbose_name?related__search=findme