In this work, we consider elliptical random fields. We propose some spatial quantile predictions at one site given observations at some other locations. To this aim, we first give exact expressions for conditional quantiles, and discuss problems that occur for computing these values. A first affine regression quantile predictor is detailed, an explicit formula is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this regression quantile is shown to be very poor for extremal quantile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional quantile. Through numerical illustrations, the study shows that Quantile Regression may perform poorly when one leaves the usual Gaussian random field setting, justifying the use of proposed extremal quantile predictions.