This is basically the whole of statistics.
Given you simulated the data from a quadratic with a specified random noise, ideally you'd get that back as your "expression of the relationship between y and x", and your significance would be in the uncertainty in the parameters of the quadratic and the residual noise estimate, using standard likelihood-based techniques.
From real data you don't know what generated it, so you make a guess based on exploratory analysis or first principles given the science or a hypothesis you want to test. Looking at your x-y plot, I'd go "hmm looks quadratic" and fit a quadratic. My first assumption would also be uncorrelated independent errors of equal magnitude.
Once fitted there's "goodness of fit" methods to tell you where your model assumptions go wrong. You might discover your errors are spatially correlated, or are much larger in one area than another, in which case you have to reformulate your model with these discoveries and fit again.
So the short answer is "statistics". Propose a model for the mean relationship, (ooh, quadratic...), model for the error (ooh, uncorrelated independent errors), then fit, refine, until you can't disprove yourself any more. Goes for spatial data, longitudinal data, experimental data - anything. Its statistics.
Is that too broad? I don't want to say "fit a polynomial model using
glm(y~Poly(x, 2)) if your real data doesn't look like this. No, you need to go through the statistical process as outlined above.