If a pawn move is absolutely forced, that usually means there's a serious threat that needs to be defended (e.g, loss of significant material, checkmate threat). Commonly, you can provoke pawn moves by piling up pressure on weak pawns, attacking the king or pinning a knight with your bishop. Basically you provoke pawn moves by creating threats; the pawn moves to defend the threat.
As for weak squares, it's only weak if you can use that square to generate new threats, or sometimes to create a defensive blockade. If not then its just a harmless hole in the position (harmless can still be useful though, sometimes your pieces just need a safe square to go to). So before trying to provoke a pawn move to weaken a square, you should also think about what you can then accomplish with that square. If you can't use it, it's not "weak".