Is there an advantage for owning enough of a dividend stock to purchase an additional share a quarter?

Not really. The return will be the same whether you own 60 shares or 120 shares. You also need to consider the wisdom of buying a share at a time. Even at a discount broker it would take you years to make back the commission at $1.50 a year.

Certainly investing more means it compounds starting at a higher dollar amount per quarter. So dollars compound faster but the rate is the same.

  As mentioned, if you  get charged a fee for each "trade" for a share of stock it is a bad idea.

Some companies have dividend reinvestment programs and they issue stock as a dividend (no fee) that gets credited to your account rather than you getting the cash and buying a "share".

I am not sure if you have to buy the stock directly from the company first or not. But google "Dividend reinvestment programs" or something like that you will find a lot of info on that.

The benefit of reinvesting dividends is you are Dollar Cost Averaging.  But I believe these days it is not like the old days when you could stick a stock in your safety deposit box for 50 years, live off the dividends, and die a millionaire.    

And realize that reinvesting dividends in more stock is a negative cash flow. You will owe tax on dividends but not get the cash.

Personally, I started out reinvesting all returns, automatically. I later realized I wanted to make the decision when it is best to invest or WHAT I wanted to invest in at the time. Things change with companies and the economy. So now all goes in to my MM account and I reinvest at will.

It compounds the same, as long as I reinvest it and don't spend it. :-)