My SGIII phone has an issue with the main board and will not start at all. Can I desolder the internal memory and move it to a Samsung Galaxy SII?
Or is there any other way to retrieve the information from the internal memory?
Firstly, this looks more like an S4 than an S III.
Memory chip pin-out and placement are heavily device-specific. What's more, older devices such as S II have distinct (individually packaged) SoC and eMMC chips, while all newer ones like S4 use chip-on-chip packaging (e.g. ePOP, as demonstrated in this previous answer I wrote - could also serve as a reference). This denies you any possibilities of transferring the chip itself.
For other means of recovery, @beeshyams already covered it well.
I have never done this and this answer is based on this XDA thread -Soldering? Move internal memory chip which can give you some pointers
EMMC is a BGA chip, which means you require specialised equipment and expertise to deal with it
IMEI information is stored in EMMC and CPU for Samsung phones. If you simply remove this chip and place in another device, IMEI mismatch will prevent radio (calling, SMS) to work, but you may be able to recover data stored
Encryption is the biggest hurdle (if your device was encrypted), even if you tackle the hardware portion. Your encryption keys are tied to the device and even if you remember the password on the old device and use the same, you may not be able to access data
Using JTAG , it may be possible to revive or locate and repair the fault. Again, buying JTAG, manipulating it's controller, transferring information etc is again for a professional or a serious hobbyist (I did try to learn this when I had a Note 2 to fix battery draining motherboard and found it too involved, though I didn't need to spend anything)
Bottom line - it's best dealt with repair or data recovery professionals and not something trivial . But then, if you are upto to it, go ahead and share your experience here for the benefit of community :)
Edit Google will tell you that both chips are different. For S3 it is
KMVTU000LM-B50 and S2
Establish if they can be interchanged? (Not within the ambit of this site) and then proceed. There can be many variations, footprint of the chip, pin connections, architecture, voltages etc which could prevent you from doing what you want