how to test endogeneity in a cross section data? Maybe it is a silly question but I really do not know much about econometrics. I am trying to look at the relationship between environmental quality and final consumption expenditure. For this, I look only at the cross-section data for Environmental Performance Index (EPI) for environmental quality and final consumption expenditure data from World Bank Database for 2010. I write down an equation of that kind : 

$$log(C)=\alpha+\beta_{1}log(S)+\beta_{2}(log(S))^{2}+\epsilon$$

where $C$ and $S$ hold for the consumption and environmental quality. 

I wonder how I can analyze for the endogeneity issues between consumption and environmental quality. Is it possible to talk about the endogeneity on cross-section data?
Endogeneity can rarely, if ever, be tested.  It can certainly exist in cross-sectional data.  

Running a presumably OLS regression without a theoretical model will just give you a measure of the co-movement of the two variables. Since no other variables are included, what you are estimating is $E(C \mid S)$, i.e. the conditional expected value of Consumption if no other information is used except for the environmental index. Including the square attempts to capture the case of a non-linear conditional expectation. 

This is more of a "prediction" framework (although not forecasting), rather than an analytical / causal framework.

Informally, you can argue for the existence of endogeneity if either there exists other variables that are correlated both with consumption and the environmental index and/or if their relation is an equilibirum one of co-determination. 

Accepting that endogeneity exists, does not invalidate your regression specification. It just says that the coefficients you are estimating may not be what you would want to estimate:
Here for example

$$\beta_1 + 2\beta_2 = \frac {\partial  E(C \mid S)}{\partial S}$$

But maybe you are interested in how "consumption changes" when the environmental index changes when other influences on consumption are taken structurally into account. This essentially implies that you would want a more rich right-hand side in your specification. 

Note: "Take structurally into account" is not the same as saying "all else held constant". If you estimated a regression with more explanatory variables, the betas themselves would change, so the marginal effect of the index in that alternative scenario would be different even if we examined under the "all else held constant" assumption.