As Belisarius and Daniel Lichtblau commented the following functions can be used for your goal.

Conditions that let the equation hold for all x:

Resolve[ ForAll[x, a x^2 + b x + c > 0] ]

(b | c) ∈ Reals && ((a == 0 && b == 0 && c > 0) || (a >= 0 && b == 0 && c > 0 && -b^2 + 4 a c > 0) || (a > 0 && -b^2 + 4 a c > 0))

Conditions that let it hold for at least one x:

Resolve[Exists[x, a x^2 + b x + c > 0]]

(b | c) ∈ Reals && (a > 0 || (a == 0 && b != 0) || (a == 0 && c > 0) || (a < 0 && -b^2 + 4 a c < 0))

SolveAlways can be used too, but it works on equations only, not inequalities.

SolveAlways[a x^2 + b x + c == 0, x]

{{a -> 0, b -> 0, c -> 0}}

As Belisarius and Daniel Lichtblau commented the following functions can be used for your goal.

Conditions that let the equation hold for all x:

Resolve[ ForAll[x, a x^2 + b x + c > 0] ]

(b | c) ∈ Reals && ((a == 0 && b == 0 && c > 0) || (a >= 0 && b == 0 && c > 0 && -b^2 + 4 a c > 0) || (a > 0 && -b^2 + 4 a c > 0))

Conditions that let it hold for at least one x:

Resolve[Exists[x, a x^2 + b x + c > 0]]

(b | c) ∈ Reals && (a > 0 || (a == 0 && b != 0) || (a == 0 && c > 0) || (a < 0 && -b^2 + 4 a c < 0))

SolveAlways can be used too, but it works on equations only, not inequalities.

SolveAlways[a x^2 + b x + c == 0, x]

{{a -> 0, b -> 0, c -> 0}}