There was an A-REIT/property trust that paid a quarterly distribution, GPT or the General Property Trust, but they are moving to twice yearly distributions like everyone else. A-REITs have distributions as opposed to dividends, same payout style without the benefit of franking credits.
Dividends are paid by all the banks, the Wesfarmers/Woolworths duopoly, Cochlear, Telstra, industrials, major miners and most mid cap miners. Franking at various rates depending on various factors. Distributions are paid by pretty much all the major A-REITS.
Well done for having an interest but keep learning some more about securities before you make some decisions. Know something about their business models, earnings, profitability, dividend yields. If you haven't done any reading yet, when I started out I found (don't laugh) the guide: Share Investing for Dummies a helpful start. Perhaps maybe look up the tutorial videos the ASX put on their website and youtube.