Before I give my guess, let's see what we have so far:
Apart from the picture links in the question, we also have this link Grace Hopper demonstrating COBOL at Programming Department (Thank you @uoho).
Most links places the picture as circa 1960 or 1957 and several also mentions COBOL.
From the WW2DB link in the question, we can see that in this time frame Grace Hopper was working for Univac. The link above even places the picture at "Programming Department", the picture originated from Sperry Rand (Corporation), Univac Division and dates it as 1957-06, ie June 1957.
The date is still a small problem. Even if I believe Hagley Digital Archives more than Wikipedia (Museum Curators normally don't give exact date without good backup evidence) it was impossible to have a COBOL class at that date, since COBOL was designed 1959 Wikipedia. So, either is the date wrong, or the COBOL reference a bit wrong. (It could have been during the work that led up to COBOL).
All of the people in the picture are also in some other pictures from Hagley Digital Archives (you might need to search around a bit). This makes me think that all 4 people were at the time part of Grace Hoppers team at Sperry Univac.
So, my best guess (until any more proof gets up) is that it was taken at Sperry Univac, at their "school room". Time somewhere between 1957 and 1960.
DrSheldon updates: This answer is correct. It was taken in June 1957 at Sperry-UNIVAC.
The original physical photograph is in possession at the Hagley Museum in the Sperry-UNIVAC collection, box 86, folder 4. It is an 8 x 10 in physical photograph., and lacks the Smithsonian watermark. (The Hagley is a Smithsonian affiliate, which is perhaps how the Smithsonian acquired a digital copy.) The Hagley documents the date as
1957-06, which is June 1957.
I do agree with UncleBod that the Hagley title for the photograph ("Grace Hopper demonstrating COBOL at Programming Department") does not fit with the timeline of COBOL. But all of the other details are consistent.
Three of the four persons (Hopper, Rothberg, and Cropper) appear in another Sperry-UNIVAC photo. It is also dated 1957, but seems to be a different room than the question photo:
Left to right: William Finley, Norman Rothberg, James McGarvey, Mary Lou Greene, Grace Hopper, and Donald Cropper.
The remaining person (Krishnan) appears with Hopper in a similar photo:
Left to right: K.C. Krishnan, Laura Yu, Donald Sullivan, George Schmalen Berger, Frederick Tweed, Grace Hopper, and Frank Lagan.
It's also worth mentioning that Hopper appears at her Sperry-UNIVAC desk with the same (or similar) clothing to the question's picture:
All of the people (including Hopper) appear to be Sperry-UNIVAC employees. Other than Hopper, none appeared at the May 1959 CODASYL meeting which drafted COBOL. It is therefore reasonable to conclude that the picture was taken at Sperry-UNIVAC.