`PossibleZeroQ` will numerically check for zero if standard transformations do not work.  It, being numerical, is not foolproof, and so it is not a complete rigorous method.  For that reason, @Artes' method is superior in this case.
    
    Simplify[
     1/12 (\[Pi]^2 + 3 ArcCosh[7/2] ArcCsch[2] - 
        12 ArcTanh[1/Sqrt[5]]^2),
     TransformationFunctions -> {Automatic, # /. _?PossibleZeroQ :> 0 &}]
    
>     `Simplify::ztest1`: Unable to decide whether numeric quantity `-3 (-Log[1+Times[<<2>>]]+Log[1+1/Sqrt[5]])^2+3 Log[1/2+Sqrt[5]/2] Log[7/2+(3 Sqrt[5])/2]` is equal to zero. Assuming it is.
    
    (*  \[Pi]^2/12  *)

`Simplify` caches results, so the message only appears on first execution.