To my surprise not even every online shop that offers picture frames, sells them in a square size. And some (like IKEA) have their own formats instead of standard sizes. But still, there are plenty of square sized frames out there.
To make it easier for you, I compiled a list with both plastic and wooden frames. They range from very cheap (3.30 € per frame) to absolutely okay (12.40 € per frame).

The easiest option to place such a group of four is a picture bar. You can buy one at IKEA or at H&M Home for example. Just place the frames aside on the rail or mix them with larger frames and other photos.
Hanging four of these tiny frames in two columns and two rows looks quite interesting, too, I think. The four frames form a unit (becoming a larger square).
Mounting picture frames can be a pain, but at these small sizes, it's a breeze. The frames just weigh 200 - 400g and only need one hanger in the middle. So, one nail in your wall for each frame will be enough. No screws needed.
Hanging with a template
However, marking and aligning can still be very annoying. I had to make my experiences here in the past. It is nerve-friendly if you simply use a template. Hold it against the wall, align it with a spirit level and hammer four nails into the marked points. Then tear off the template and hang up the frames.
I have prepared a template for you that you can download and print out. Please print out the template in 100% size. PDF readers often scale the pdfs a bit down which is not what you want in this case.
I placed the dots 18.4cm apart from another which should result in a distance of 1 - 2 cm between the frames (depending on the width of your frame).
If you prefer more distance between the frames, fold the paper at the dotted line. This will result in a square piece of paper with a side length of 21cm. Now simply put one nail at each corner of the square paper.
Choice of glass
In May I already wrote an article, mentioning different types of glass. When you're hanging small frames, the choice of glass is not of highest importance, I would say.
Both normal glass and float glass work well and I can totally recommend these traditinal types of glass. But even museum glass (e.g. UV70) becomes quite affordable at these small sizes and could be considered if you want to spoil yourself.

You can also go with acrylic glass, if you don't forget to remove the protective film before hanging it on your wall (again, speaking from experience).
Tip: Testing out frames, I had an idea: I would consider removing the glass entirely (and putting it behind the passepartout, in case you will need it in the future). With a good passepartout and a print on Hahnemühle Photo Rag Baryta, it will look fantastic framed without glass.

Here is a compilation of frames I can recommend. I sorted them by price from cheap to more expensive. Personally I would choose a frame in white or black or in the color of your floor, but of course, this is up to you. Shipping costs vary, so you might want to keep an eye on those as well, especially if you want to order four frames.
Picture frames in 15x15cm
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Puro plastic picture frame (e.g. white or black)
3.30 €, normal glass (shipping 6.95 €) -
Walther plastic picture frame (e.g. white or black)
3.35 €, normal glass (shipping 4.95 €) -
Edsbyn wooden picture frame (white)
7.03 €, float glass (shipping 5.90 €) -
Edsbyn wooden picture frame (black)
7.03 €, float glass (shipping 5.90 €) -
Sunny mdf wooden picture frame (e.g. black or white)
8.03 €, acrylic glass (shipping 4.90 €) -
Uno solid wooden frame (e.g. black or white)
8.75 €, float glass (shipping 5.90 €) - Lund solid wooden frame (e.g. black or white) 12.40 €, museum glass UV70 (shipping 6.95 €)
The cheap ones:
Wooden frames:
With museum glass:
My personal preference would be Uno frame (black, white or another wood finish depending on your taste) with the glass entirely removed when hanging.
