Unlikely, or it depends if you can figure it out yourself if you can or can not, as it may or may not be possible for various reasons.
The other board looks like has a programmable microcontroller, so it is scanning the keypad matrix. The connector may have 10 pins instead of 7 for various reasons. The matrix may or may not be arranged as a 3x4 matrix. The other pins might be unused or used for something, like ground, chassis, earth, or backlight or whatever. It is unknown until you figure out how the 10 pins are used.
Even if it just a 3x4 matrix, you don't know how it is scanned until you test it. It might be a part of a larger matrix. You also don't know if the matrix is scanned with 5V or 3.3V or some other voltage, until you figure it out.
Even if you figure it out, the speed how the matrix is scanned is also unknown, so it may be difficult to write software which reacts to correct scan outputs fast enough to simulate a button push. Might be especially hard with Raspberry Pi since it does not run a real time OS. Might be possible with Arduino if most of the time is spent responding to the scanned matrix pins, instead of other stuff.