lst = {{4, 1}, {9, 2}, {16, 3}, {25, 4}};
lst[[All, 1]] = Sqrt@lst[[All, 1]];
lst

{{2, 1}, {3, 2}, {4, 3}, {5, 4}}

And

lst = {{4, 1}, {9, 2}, {16, 3}, {25, 4}};
lst = ReplacePart[lst, {a_, 1} :> Sqrt[lst[[a, 1]]]]

{{2, 1}, {3, 2}, {4, 3}, {5, 4}}

lst = {{4, 1}, {9, 2}, {16, 3}, {25, 4}};
lst[[All, 1]] = Sqrt@lst[[All, 1]];
lst

{{2, 1}, {3, 2}, {4, 3}, {5, 4}}

Also

lst = {{4, 1}, {9, 2}, {16, 3}, {25, 4}};
Transpose[{Sqrt[#[[1]]], #[[2]]}&@Transpose[#]]&@lst

{{2, 1}, {3, 2}, {4, 3}, {5, 4}}

And

lst = {{4, 1}, {9, 2}, {16, 3}, {25, 4}};
lst = ReplacePart[lst, {a_, 1} :> Sqrt[lst[[a, 1]]]]

{{2, 1}, {3, 2}, {4, 3}, {5, 4}}