I was curious so I did a little rough math and I came up with around $420 for a 24K gold u.s quarter.
How far off the mark am I?
Taking my own measurements of a quarter (D=0.955" x H=0.067"), the density of gold (19.3 g/cc) and the current price of gold per gram ($25.19) I get a slightly lower figure, $382.35. This represents an upper bound, since a quarter is not a true cylinder, since much of the flat surfaces are below the level of the rim.
I believe a quarter weighs approximately 1/3 of an ounce, so if gold is $783 (latest call) per that would make a solid gold quarter worth about $258.
I thought he meant the weight of a quarter, not the relative size.
The density of gold is higher than the alloys which comprise a quarter. Thus, a quarter of the same physical dimensions would weigh more.
A gold coin that weighed the same as a regular quarter would be a much smaller coin, due to the differing densities of the metals.
*** Ponder
Now wait a minute; is this a trick question? A quarter is worth .25, so a quarter made of gold would be worth .25, but it would only weigh 0.0089 grams. 
According to Wikipedia, the Height is 0.069", which is slightly higher than your measurement.
Using the figures you provided, the upper bound would be ~$394.
Roughly 400 buckaroos. Way too much to carry a small coin purse full of gold coins instead of a wallet.
Silver quarters would cost less than $3 each which is too little to be convenient.
And this, ladies and gents, is why we are stuck with “paper” bills and plastic cards. We need a new, shiny metal worth…say…$15 / quarter and…umm… $2 / dime.
We’re only interested in the physical shape of the quarter, not its 25cent face value.
A US quarter is about the same diameter/thickness as a quarter ounce US gold eagle. The quarter is about 10% larger in diameter. A gold quarter, if it were pure, would have about $240 in intrinsic gold value. I’m just roughing it out here.
This assumes gold is $783. US, which is what it closed on Friday.
All of which should put loot in D&D games into much clearer perspective: Even the copper piece is worth significantly more than a dollar, and there aren’t even any denominations smaller than that.
Just to keep it honest, even though it isn’t the question, a US circulated quarter made between 1965 and today weighs 5.67 Grams or 0.18 ounce, according to the Coin World guide to US coins.
Due to variable relief issues wouldn’t the most precise thing to do be to get the exact weight of a current quarter then determine the density of the metal and adjust the ratios as needed.
Quarter - Cupro-Nickel = 8.33% Ni - 81.67% Cu -
Total weight = 5.670 grams
I’m not quite sure how to do the math with a blend like this but I imagine someone here should be able to.
Density at room temp of (via wiki):
Gold - 19.3 g·cm−3
Copper - 8.96 g·cm−3
Nickle -8.908 g·cm−3
Those are 22 karat, FTR.
If we estimate the average density of the cupro-nickle blend to be approximately 8.945 g per cubic cm and the actual weight of the quarter is 5.67 g / 8.945 = a quarter being .663 of a cubic cm. Thus a quarter would contain 19.3/.663 = 12.8 grams of gold x 25.10 per gram = 322.43. I should mention that my 8.945 density blend estimate is a guess based on estimating the weight and ratios. If someone has the mad math skillz to determine the exact density of the blend that number will potentially swing 10 - 15 one way or the other.
Well, I’m not sure if it’s usual to measure the metal content of alloys be volume or weight, but here are my numbers:
Percents by volume:
Density of alloy: 8.956 g/cm^3
Volume of quarter: .566 cm^3
Weight of gold quarter: 10.9197 g
Value of gold quarter at $783/oz: $274.89
Percents by weight:
Volume of quarter: .633 cm^3
Weight of gold quarter: 12.2192 g
Value of gold quarter at $783/oz: $307.61
At least, provided my math is good. Given that gold prices change, I’d say roughly $300 would be a decent guess.