Linear regression is meant for linear relationships right, so, if I don't trust linear correlation and want to find out if random variables $y$ and $x$ have a non-linear relationship, can't I just regress $y$ against $x$ without intercept like so, $y = \beta x + \epsilon$, and then use a QQ plot of the residuals to see that a divergence from the 45-degree line indicates non-linearity. Otherwise, maybe some other related post-regression test?
No, the QQ plot doesn't tell you about the relationship between y and x's, it tells you about the distribution of the residuals (which should reflect the errors if the model is otherwise correct, if the QQ plot doesn't look fairly close to linear the residuals are not close to what you'd expect if the errors were normally distributed).
Residual plot(s) tell you about non-linearity is the relationship between y and the corresponding x variable(s).
Here's what you could see for an example set of data with mild non-linearity in it.
Here there is a bit of non-linearity (because I put it in the data), but it is not totally obvious in the plot of y vs x.
If the linear model were correct the residuals should appear to be randomly scattered above and below 0 at each x-value.
It is not the case here -- in the plot of residuals vs x you can see the curvature clearly. I marked in a quadratic curve but you'd more typically look at a smooth fit to the residuals for such a purpose.
The QQ plot looks linear here but it is not readily interpretable because of the issue in the residual plot.
(There are better things to plot than raw residuals but let's get the more basic concepts clear to begin with.)
Before even getting to regression modelling, if you just have two scalar variables then you should start with a scatterplot of $x$ and $y$. That is likely to immediately tell you if they have a "linear" (or perhaps "affine") relationship. It is of course possible to follow this up with a formal regression model to test for non-linearity between the two variables; one way to do this is to fit a polynomial regression and test whether the higher-order coefficients (and constant term) are all equal to zero.