Squaring is raising a number to the power of two, an exponential operation. Growing in proportion to a power of two could be reasonably described as a quadratic rate of growth.

The pedantry obviously isn't relevant to the discussion, but just so you're clear why everyone is calling you wrong: "exponential growth" refers to a function of x that goes as C^x for some constant C, not one that goes as x^C. The latter is "polynomial" (of which "quadratic" is the term used when C is 2).

Exponential is a^x. Square is polynomial, i.e. x^a.