Interpretation of a differenced regression
If the levels specification is deemed acceptable,

$$y_t = \beta_0 + \beta_1 x_t + \beta_2 x_t^2$$

then it follows that the first-difference specification should not include a constant term in order to be methodologically consistent,

$$\Delta y_t = \beta_1\Delta x_t + \beta_2\Delta x_t^2$$

since $\Delta \beta_0 = \beta_0 -\beta_0 = 0$.

I suggest you run the specification of first-differences without intercept and see what happens.

Note: the slope coefficient estimates you should obtain in the first-difference specification without intercept should be close to the corresponding estimates you obtained in the levels-estimation. Otherwise, the maintained hypothesis of constant slope coefficients becomes questionable, or the model has other misspecification issues. 

(this is not to be confused with the case where we run a regression without constant term but with the dependent variable and the regressors centered on their sample means -there we would obtain the exact same slope estimates with the levels specification that includes a constant term, as a matter of algebraic property of least-squares estimation).