Please note that these are not mutually exclusive alternatives, since you can also generate prediction intervals using quantile regression.
In general, whatever you choose, you want the prediction intervals, because you are interested in the error of the prediction you are making rather then in the error of the estimated relationship.
Regarding unequal variances, I would go for a linear regression model rather than for quantile regression, but mostly because I know the former better. A gradually increasing variance is not a great issue and you can work with that in linear regression model. Specifically, I would go for robust regression (e.g. rlm from MASS R package), but there are also alternatives using generalized linear models (glm's), like gls from the nlme R package.
However, I would first ask myself whether the data should not be transformed. The behavior that you are describing is typical for data that needs to be log-log transformed (you logarithmize both x and y) - for example, correlation between colony forming units (CFU) of bacteria and the measured raw fluorescence intensity.