Is it possible to use cross-validation to estimate the reliability of a specific predictor? I have a big dataset with many predictors. I would like to know the quality of these predictors using cross-validation. However, what I am finding is more so methods that test the reliability of the model as a whole.
Is it possible to check the reliability of specific predictors?

Generally, cross-validation is used to assess predictive value. From that perspective, you could fit an otherwise identical model with and without a variable and assess the change in the root mean squared error of the prediction (RMSEP).
Many researchers use "reliability" as a synonym for significant.  I gather that is your intention here.  Without intending any disrespect, I think this is a poor usage.   If you haven't done any data snooping / p-hacking, there is no reason to be concerned that many variables are significant at $\alpha = 0.05$ when you have many data.  If you want, you could always use a more stringent alpha to reduce the risk of type I errors when you have many data.
