The generalized quadrangle Q(4,q) arising from the parabolic quadric in PG(4,q) always has an ovoid. It is not known whether a minimal blocking set of size smaller than q2 + q (which is not an ovoid) exists in Q(4,q),  q odd. We present results on smallest blocking sets in Q(4,q),  q odd, obtained by a computer search. For q = 5,7,9,11 we found minimal blocking sets of size q2 + q - 2 and we discuss their structure. By an exhaustive search we excluded the existence of a minimal blocking set of size q2 + 3 in Q(4,7).