One q-analogue of permutations as bijections is the set of invertible matrices over a finite field. We study the general problem of counting matrices over finite fields with certain rank and with support that avoids a set S of the entries. These matrices are a q-analogue of permutations with restricted positions and it is known that for general sets these numbers are not polynomials in q (Stembridge 98). We give polynomial q-analogues of derangements (counting invertible matrices with zero diagonal) answering a question of Stanley, and of fixed point free involutions (counting invertible symmetric matrices with zero diagonal in odd characteristic). We then move to the issue of polynomiality starting from Haglund's relation to the Garsia and Remmel q-rook numbers when S is a Ferrers board, and give evidence of reciprocity and polynomial behavior for other families of sets like skew shapes. This is joint work with Joel Lewis, Ricky Liu, Greta Panova, Steven Sam, and Yan Zhang.