Probably not, the pause button isn't a readable input button like the buttons on the joypad are. It fires an interrupt (NMI) when pressed, which jumps the code execution to a part of the rom where the game has its pause-related code. It's also impossible to detect the holding or the release of the pause button because of this, only the initial triggering.
Is it the following mod you are planning to use?
https://www.smspower.org/Development/JoypadPauseButtonIt seems to wire a pushbutton on the joypad to an impossible button combination (up+down or left+right) and by also adding a small circuit to the joystick port in the console so that it fires an NMI when that combination is made, it will act just like the pause button. So yes, both joypad and console must be modded with this method. You could also just add a separate wire for the pushbutton wired directly to the NMI line, but in that case you would also need to add a separate connector on the console for the pushbutton's wire.
On the bottom of the page they warns that heavily used joypads may allow you to push in opposite directions and accidentally trigger a pause that way.