The new Hamad International airport in Doha isn't quite finished yet, so things may change over the next year or so. As an example, in November 2014 the Qatar Airways First Class lounge isn't yet open (not that it'll matter much for your situation, but still...!)
If you are flying Qatar Airways in Business Class, then you'll have access to the very impressive looking Al Mourjan lounge (recent review with photos). If you're flying Qatar Airways First Class, for now you'll also use the Al Mourjan lounge, but at some point you'll switch to the Al Safwa First Class Lounge when it opens.
Both of those are class of service based. Frequent Flier cards don't get you in, credit cards don't get you in, only a boarding pass in a premium cabin on Qatar Airways (or OneWorld partner) will do so.
For those who have a OneWorld frequent flier status which permits lounge access (Emerald or Sapphire) flying in Economy, there are two other lounges available. There's the 'Qatar Airways First Class Lounge' or 'Al Maha' for OneWorld Emeralds (which isn't where Qatar Airways First Class ticket holders go), while both Emerald and Sapphire travellers can make use of the 'Qatar Airways Business Class Lounge' (ditto)
Finally, there's the Oryx Lounge. This is available to OneWorld Emerald and Sapphire card holders if they want, but as detailed on the airport's website
Economy passengers will be able to gain access to the lounge for a nominal fee of QAR 140
I can't see that lounge on the priority pass website, nor on the American Express Platinum chargecard lounge finder (login required), so I don't believe that a high end credit card will get you in, sorry. So, looks like your only option is to pony up the 140 riyals (~USD 40) to pay your way in.
(I have seen suggestions online that when the other lounges are complete, the paying Oryx Lounge will start accepting high end credit cards and things like priority pass, but that isn't certain and there's no start date I can find yet for it, so I wouldn't count on it)
Note that there is also an airside Airport Hotel, but I can't find any prices so assume they're a lot more than the lounge. Unless you both want to sleep and can't get a visa on arrival to enter Qatar (you can get some cheap-ish hotels in Doha if you look), you probably won't want to be paying the higher prices for that.