Summary: A two-character set in \(\mathrm{PG}(r,q)\) is a set \(\mathcal{X}\) of points with the property that the intersection number with any hyperplane only takes two values. A projective Paley set of \(\mathrm{PG}(2n - 1, q)\), \(q\) odd, is a subset \(\mathcal{X}\) of points such that every hyperplane of \(\mathrm{PG}(2n - 1, q)\) meets \(\mathcal{X}\) in either \((q^n + 1)(q^{n-1} - 1)/2(q - 1)\) or \((q^n - 1)(q^{n-1}+1)/2(q - 1)\) points. A quasi-quadric in \(\mathrm{PG}(2n - 1, q)\) is a two-character set that has the same size and the same intersection numbers with respect to hyperplanes as a nondegenerate quadric. Here we construct projective Paley sets of \(\mathrm{PG}(3, q)\) left invariant by a cyclic group of order \(q^2 + 1\) and of \(\mathrm{PG}(5, q)\) admitting \(\mathrm{PSL}(2, q^2)\) as an automorphism group. Also infinite families of quasi-quadrics of \(\mathrm{PG}(5, q)\) are provided.