The density of gold is 161.2 lb/ft^3. Pathfinder gp are 50 to the pound, so their volume is 0.000124 cubic feet each. The portable hole has a volume of (pi/4)*6^2*10 = 282.7 cubic feet. That fits 2,278,349 gp. (Now you just pretend that all the other metals have the same density as gold.)
But Fuzzy is assuming perfect packing isn't he? Coins are cylinders and there will be <quick google, let's pretend this is packing circles in a plane...> no better than 90.69% efficiency.
So 2,278,349 * 0.9069 = 2,066,234gp.
I found a site that calculates 12,000 coins per cubic foot: https://olddungeonmaster.wordpress.com/2012/06/09/coins/
Which would mean roughly 3.4 million coins, which is pretty close to Fuzzy's answer. However, I don't think it's right. First, the density of gold is almost ten times what Fuzzy said it is: 1206 pounds per cubic foot. Secondly, that site I linked to is using coins that are larger in every dimension than US quarters, and 50 gold coins that big would weigh far more than one pound.
On the other hand as Ginoa says, you can't just assume that the hole is literally solidly packed with coins, there's going to be some amount of wasted air space between the coins. (Around 10% wasted space if they were carefully stacked (which wouldn't be happening with the shovels you mentioned), or maybe around 35% wasted space randomly shoveled in.) So, with a usable volume of 282.7-(35% of that)=134 cubic feet and coin volume .0000166 cubic feet...
More like 8.1 million coins. Maybe as high as 9.3 million if the coins were packed fairly closely but not perfectly, maybe as low as 6.2 million if they were pretty loose. Or, as high as about 11.2 million if they somehow managed to get them stacked almost perfectly.
As a side note, if the coins were pure gold, they'd be much smaller than a US quarter to have 50 of them equal one pound, although most of the calculations in these posts worked out the coin size by metal density as opposed to assuming a particular size to start with. http://imgur.com/NZbCcfp
That's what I get for being lazy :-( Should have checked the page I googled for the density more carefully, and then solved the packing problem!
But there do exist square coins :-P
DnD coins are a tad over an inch in diameter.
Approximately 12,000 coins loosely piled per cubic foot, according to a sidebar in the draconomicon. So (pi*r^2)*h -> 3.14*9*10 -> 270ish cubic feet. 270*12,000=3,240,000 coins shoveled in.
Wasn't this a query in an old Dragon magazine, from which I believe the answer was a million. However the coins may have been a different size.
Although remember gold coins are probably adulterated, lest they be too malleable & easily milled. Thus volume rather than density/mass would be the preferred method of calculation (as Darigaaz shows).