I believe that early immigration stuff needs to found at state level i.e. in this case Queensland State Government (https://www.qld.gov.au/recreation/arts/heritage/archives ). I know that the indices are online, but I think that you might need to contact them if you want a copy of the actual passenger lists. (Note there seems to be a problem searching the assisted immigration db at the moment, but you can use one of the alphabetic indices that will display on the right hand of the screen)
If you know any passenger names, you could try ancestry and/or findmypast. There is a chance that they have the passenger lists as well.
PS I have just realised that they have added a link in the indices to a copy of the actual passenger list!!