a = {{1, 2}, {2, 3}};
b = {{1, 3}, {3, 3}, {4, 1}};
function[x__] := x//Intersection
function[a[[;;,1]], b[[;;,1]]]

a = {{1, 2}, {2, 3}};
b = {{1, 3}, {3, 3}, {4, 1}};
function[x__] := x[[;;,1]]//Intersection
function[a, b]
Part::pkspec1: The expression {{1,3},{3,3},{4,1}} cannot be used as a part specification.
Part::pkspec1: The expression {{1,2},{2,3}} cannot be used as a part specification.

function[a]
{1,2}